Mechanisms for Constructing Spline Surfaces to Provide Inter-Surface Continuity

ABSTRACT

A mechanism is disclosed for reconstructing trimmed surfaces whose underlying spline surfaces intersect in model space, so that the reconstructed version of each original trimmed surface is geometrically close to the original trimmed surface, and so that the boundary of each respective reconstructed version includes a model space trim curve that approximates the geometric intersection of the underlying spline surfaces. Thus, the reconstructed versions will meet in a continuous fashion along the model space curve. The mechanism may operate on already trimmed surfaces such as may be available in a boundary representation object model, or, on spline surfaces that are to be trimmed, e.g., as part of a Boolean operation in a computer-aided design system. The ability to create objects with surface-surface intersections that are free of gaps liberates a whole host of downstream industries to perform their respective applications without the burdensome labor of gap repair, and thus, multiplies the efficacy of those industries.

PRIORITY CLAIM

This application claims benefit of priority to Application No. 62/295,892 titled “Construction of Spline Surfaces to Provide Inter-Surface Continuity”, filed on Feb. 16, 2016, and Application No. 62/417,781 titled “Mechanisms for Constructing Spline Surfaces to Provide Inter-Surface Continuity”, filed on Nov. 4, 2016, and which are both hereby incorporated by reference as though fully and completely set forth herein.

ACKNOWLEDGEMENT OF GOVERNMENT SUPPORT

This invention was made with government support under Grant No. N00014-08-1-0992 awarded by the Office of Naval Research. The government has certain rights in the invention.

FIELD OF THE INVENTION

The present invention relates generally to the field of computer-aided design (CAD), and more particularly, to mechanisms for constructing (including reconstructing) tensor product spline surfaces to address the trim problem.

DESCRIPTION OF THE RELATED ART

The computer-aided design of models for objects is fundamentally important to a variety of industries such as computer-aided engineering (CAE), computer-aided manufacturing (CAM), computer graphics and animation. For example, a model may be used as the basis of an engineering analysis, to predict the physical behavior of the object. The results of such analysis may be used in a wide variety of ways, e.g., to inform changes to the design of the object, to guide selection of material(s) for realization of the object, to determine performance limits (such as limits on temperature, vibration, pressure, shear strength, etc.), and so forth. As another example, a model may be used to direct the automated manufacturing of the object. As yet another example, a model may be used to generate an image (or a sequence of images, e.g., as part of an animation of the object). These activities may represent typical product design development steps found across industrial market verticals such as automotive, aerospace, and oil and gas.

Modern premiere CAD software applications may be built on software kernels that utilize restrictive mathematical assumptions to approximate compound geometric objects. As a result, critical information may not be explicitly modeled, forcing designers, engineers, manufacturers, animators, etc., to repair CAD models and convert them into an acceptable format, such as polygonal meshes for Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD), 3-D printing or additive manufacturing, character animation, etc. This conversion is typically an iterative process, creating substantial amounts of work for product development teams. Designers, engineers, manufacturers, animators, etc., waste countless hours manually repairing gaps in models and dealing with redundant one-way file conversion operations, causing significant productivity losses, increased time to market and user frustration and dissatisfaction. As such, improvements in the field of CAD modeling may be desirable.

SUMMARY

Embodiments are presented herein of methods, computer systems, and computer-readable memory media for constructing gapless surface models in computer-aided design (CAD) applications. In one set of embodiments, a computer-implemented method for modifying a model is implemented, wherein the method comprises performing, by the computer: storing geometric input data describing a first and second input parametric surface associated with the model, wherein the first and second input parametric surfaces are described in a first and second parameter space domain, respectively. The method may proceed by storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve approximates a geometric intersection of the first and second input surfaces. The method may proceed by reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve. The method may proceed by constructing first and second output surfaces, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve. Finally, the method may output the first and second output surfaces as modified components of the model.

In some embodiments, the first input surface may include a first tensor product spline surface; the first geometric input data may include a first pair of knot vectors and a first set of surface control points and weights; the second input surface may include a second tensor product spline surface; and the second geometric input data may include a second pair of knot vectors and a second set of surface control points and weights.

In some embodiments, the first and second tensor product spline surfaces may be NURBS surfaces (or T-Spline surfaces).

Some embodiments may comprise a pre-SSI algorithm, wherein it is assumed the user has two boundary representations (Brep, brep, B-rep or b-rep) they wish to perform a solid modeling Boolean operation on. Other embodiments may comprise a post-SSI algorithm, wherein it is assumed the user has a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. Pre-SSI input data may comprise two B-reps upon which it is desired to perform solid modeling Boolean operations. Post-SSI input data may comprise a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. For both the pre-SSI and post-SSI algorithms, geometric input data may be stored describing a first and second input surface associated with the model under consideration. Each of the first and second input surfaces may be described as parametric surfaces in a first and second respective parameter space domain.

The input for the Post-SSI algorithm may be a boundary representation created using one or more (typically many) solid modeling Boolean operations on spline surfaces. (The spline surfaces may include tensor product spline surfaces such as NURBS surfaces and/or T-Spline surfaces.) In one embodiment, the boundary representation may be provided in a computer-aided design (CAD) file. In another embodiment, the Post-SSI algorithm may operate as part of the software kernel of a CAD system, in which case the boundary representation may have a specialized internal format.

In some embodiments, the first output surface geometrically approximates a subsurface of the first input surface to within model tolerance, and the second output surface geometrically approximates a subsurface of the second input surface to within the model tolerance, wherein model tolerance may be predetermined or user-defined.

In some embodiments, the boundary portion of the first output surface patch corresponds to a boundary isoparametric curve of the first output surface patch, wherein the boundary portion of the second output surface corresponds to a boundary isocurve of the second output surface. (Note that the term “isocurve” is used as a synonym for isoparametric curve.)

In some embodiments, the method may also include: numerically computing a set of intersection points that at least approximately reside on the geometric intersection of the first input surface and the second input surface; computing geometric data (e.g., knot vector and curve control points) that specify a model space curve, approximating the intersection of the first and second input surfaces as a parametric curve, based on the set of intersection points; and storing the geometric data that specify the model space curve.

In some embodiments, the method may also include executing an engineering analysis (e.g., a physics-based analysis) based on the modified model, to obtain data predicting physical behavior of the object. (The predictive data may be used to: generate display output to a user, for visualization of the physical behavior; calculate and output a set of performance limits for a manufactured realization of the object; identify locations of likely fault(s) in a manufactured realization of the object; select material(s) to be used for manufacture of the object; direct a process of manufacturing the object; direct automatic changes to geometry of the boundary representation; etc.)

In some embodiments, the above mentioned engineering analysis may comprise an isogeometric analysis.

In some embodiments, the method may also include: after having performed said modification of the model, manufacturing (or directing a process of manufacturing) the object based on the boundary representation.

In some embodiments, the method may also include: after having performed said modification of the model, generating an image of the object based on the boundary representation and displaying the image.

In some embodiments, the method may also include: after having performed said modification of the model, generating a sequence of animation images based on the boundary representation and displaying the sequence of animation images.

In some embodiments, the action of creating the first geometric output data includes: determining a first set of isocurve data specifying a first set of isocurves of the first input surface on a region within a parametric domain of the first input surface, dividing the first set of isocurves at respective locations based on the model space curve, to obtain a second set of isocurve data specifying sub-isocurves of the respective isocurves; and computing surface control points for the first output surface patch based on control points for said portion of the model space curve and a proper subset of the second set of isocurve data. (In this context, “proper subset” means a non-empty subset that excludes the end control points nearest the model space curve.)

In some embodiments, the above described sub-isocurves are reparametrized to a common parametric interval and cross-refined to achieve a common knot vector prior to said computing the surface control points for the first output surface patch.

In some embodiments, the action of computing the surface control points for the first output surface patch comprises solving one or more linear systems of equations that relate said surface control points to the control points for said portion of the model space curve and said at least a portion of the first set of isocurve data.

In some embodiments, the method may operate in the context of surface modeling, in which a topological object model (i.e., faces, edges, and vertices) has been created to reference the surface model objects (i.e., the tensor product spline surfaces, the model space curve, and a parameter space curve per tensor product spline).

Thus, the two output surfaces may meet in a C⁰ continuous fashion along the model space trim curve, corresponding to the isoparametric curve boundaries of the surfaces, and closely respect the CAD designer's selection of original geometry for the trimmed surfaces. (The model space trim curve is associated with the topological edge between the two trimmed surfaces.) Thus, after application of this method, the above-described gaps at surface-surface intersections may no longer exist as per the user's intent. Having made the surfaces watertight, the boundary representation may be immediately ready for any of various applications such as finite element analysis (e.g., conventional analysis or isogeometric analysis), graphics rendering, animation, and manufacturing (e.g., 3D printing, such as additive manufacturing). For example, conventional mechanisms may be invoked to automatically convert the boundary representation into a continuous polygonal mesh, for any of the above listed downstream processes. As another example, the boundary representation may be subjected to an isogeometric analysis, in which the geometric basis functions, implicit in the output surfaces or refinements thereof, are used as the analysis basis functions.

Note that each of the above described output surfaces may be represented by a corresponding set of one or more output surface patches. Thus, the above stated conditions on the two output surfaces may be interpreted as conditions on the two sets of output patches. In particular, for each set of output patches, the boundary of the union of the images of the set of output patches includes the model space trim curve, and the union of the images of the set of output patches is geometrically close to the corresponding trimmed surface.

Furthermore, the above described method naturally generalizes to the reconstruction of two or more trimmed surfaces that nominally intersect along a given model space trim curve. (We use the term “nominally” because the trimmed surfaces actually meet in a non-continuous gap-replete fashion.) Thus, in some embodiments, the method may operate on the two or more trimmed surfaces to generate two or more corresponding output surfaces that meet in a C⁰ continuous fashion along the model space trim curve, and closely respect the CAD designer's selection of original geometry for the trimmed surfaces. These embodiments may be especially useful for the design of non-manifold objects, or objects including non-manifold structures. (As an example of how more than two nominally intersecting trimmed surfaces might arise, imagine the Boolean union of two arbitrary spline surfaces that intersect in model space. This operation would give four trimmed surfaces that nominally intersect along a model space trim curve.)

In prior art solid modeling technology, the geometry of the spline surfaces S₀ and S₁ is never altered when performing the Boolean operations on the spline surfaces. Unfortunately, this commitment to unaltered geometry makes it impossible for the conventional Boolean operations to create surface-surface intersections that are gap free. (Typically, surface-surface intersections exhibit numerous small scale gaps and openings, making the model non-watertight.) The Brep object resulting from a conventional Boolean operation is un-editable and static. As a result, downstream processes such as finite element analysis and 3D printing cannot be performed until the solid model is rebuilt, typically with many hours of painstaking human labor.

According to embodiments presented herein, continuity of surface-surface interface may ensure that the boundary representation model is immediately ready for downstream applications such as engineering analysis or manufacture or graphics rendering. Because there are no gaps at the surface-surface interfaces, there may be no need for gap remediation.

In some embodiments, the action of constructing the boundary representation model of the object may be performed as part of a solid modeling Boolean operation in a CAD software system on a set of two or more input surfaces, wherein each of the output surfaces corresponds to a respective one of the input surfaces. In some embodiments, the action of constructing the boundary representation may be performed internal to a CAD software application.

In some embodiments, for each pair of the input surfaces that geometrically intersect, the corresponding pair of output surfaces meet in a continuous fashion along respective isocurves.

The method for creating the output patches may do so in a watertight configuration, such that the output surfaces meet in at least a C⁰ continuous fashion along respective isocurves. This method may provide geometric as well as parametric compatibility, in which the domains of the output surfaces are created to define a previously missing single domain for the output surfaces for which desirable properties are furnished (e.g., properties such as geometric and parametric continuity, parametric structure, parametric fidelity, mesh resolution, mesh aspect ratio, mesh skew, mesh taper, etc.).

In some embodiments, the method may also include: (a) performing a computer-based engineering analysis on the object based on the boundary representation model (without any need for gap-remediation on the boundary representation model), wherein the computer-based engineering analysis calculates data representing physical behavior of the object; and (b) storing and/or displaying the data representing the physical behavior of the object. In some embodiments, the data represents one or more of: a predicted location of a fault in the object; a decision on whether the object will tolerate (or survive or endure) a user specified profile of applied force (or pressure or stress or thermal stimulus or radiation stimulus or electromagnetic stimulus); a predicted thermal profile (and/or stress profile) of an engine (or nuclear reactor) under operating conditions.

In some embodiments, the method may also include directing one or more numerically controlled machines to manufacture the object based on the boundary representation model (without any need for gap-remediation on the boundary representation model). The physical surfaces of the manufactured object meet in a continuous fashion. In some embodiments, the numerically controlled machines include one or more of the following: a numerically controlled mill, a numerically controlled lathe, a numerically controlled plasma cutter, a numerically controlled electric discharge machine, a numerically controlled fluid jet cutter, a numerically controlled drill, a numerically controlled router, a numerically controlled bending machine.

In some embodiments, the method may also include: (a) employing a 3D graphics rendering engine to generate a 3D graphical model of the object based on the boundary representation model (without any need for gap remediation on the boundary representation model); and (b) storing and/or displaying the 3D graphical model of the object (e.g., as part of a 3D animation or movie).

In some embodiments, the method may also include: (a) performing 3D scan on the boundary representation model to convert the boundary representation model into a data file for output to a 3D printer (without any need for gap remediation on the boundary representation model); and (b) transferring the data file to a 3D printer in order to print a 3D physical realization of the object. Due to the continuity of meeting between surfaces of the boundary representation, the printed physical realization will have structural integrity (and not be subject to falling apart at surface-surface interfaces).

In some embodiments, the method may also include manufacturing a portion of a body (e.g., a hood or side panel or roof section) of an automobile based on the boundary representation model of the object. Surfaces of the body portion meet in a continuous fashion.

In some embodiments, the method may also include manufacturing a portion of a body of a boat (or submarine) based on the boundary representation model of the object. Surfaces of the body portion meet in a continuous fashion.

In some embodiments, the method may also include: (a) downloading each of the output surfaces to a corresponding robotic manufacturing device; and (b) directing the robotic manufacturing devices to manufacture the respective output surfaces. Furthermore, the method may also include assembling the manufactured output surfaces to form a composite physical object.

In some embodiments, the boundary representation model represents a hydrocarbon reservoir in the earth's subsurface. In these embodiments, the method may include performing a geophysics simulation based on the boundary representation model in order to predict physical behavior (e.g., a flow field or a pressure field or a temperature-pressure field) of one or more hydrocarbons in the hydrocarbon reservoir. The method may also include determining one or more geographic locations for drilling of one or more exploration and/or production wells in the hydrocarbon reservoir. The method may also include determining a time profile for production of the one or more hydrocarbons via one or more wells based on the predicted physical behavior.

This summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, Figures, and Claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described with reference to the attached drawings in which:

FIG. 1A is an illustration of intersecting first and second spline surfaces, according to some embodiments;

FIG. 1B is an illustration of the intersecting first and second spline surfaces of FIG. 1A that have been trimmed along their intersection, according to some embodiments;

FIG. 1C is an illustration of the gaps that may occur along the trimmed intersection of FIG. 1B, according to some embodiments;

FIG. 2 is an illustration of an exemplary computer system that may be used to perform any of the method embodiments described herein, according to some embodiments;

FIG. 3 is an illustration of a user creating a CAD model of an automobile part, according to some embodiments;

FIG. 4 is an illustration of a conventional method for performing solid modeling Boolean operations, according to some embodiments;

FIG. 5 is an illustration of a conventional B-rep solid model of an actual automotive part, according to some embodiments;

FIG. 6 is an illustration of how a conventional B-rep model may introduce gaps that may hinder animation of a model, according to some embodiments;

FIG. 7 is an illustration of a conventional faceted quadrilateral mesh model as compared to the watertight B-rep model, according to some embodiments;

FIG. 8 is a graphical illustration of the watertightCAD method, according to some embodiments;

FIG. 9 is a flowchart of the watertightCAD method indicating five steps, including the three core technical steps, used to transform a traditional B-rep solid model into a watertight model, according to some embodiments;

FIG. 10 is a flow chart illustrating the algorithmic steps involved in the watertightCAD methodology that may be performed depending on whether pre-SSI or post-SSI input data is received, according to some embodiments;

FIG. 11 is a flow chart illustrating the steps of the preprocessing algorithm when pre-SSI input data is received, according to some embodiments;

FIG. 12 is a flow chart illustrating the steps of the preprocessing algorithm when post-SSI input data is received, according to some embodiments;

FIG. 13 is a flowchart diagram illustrating the steps involved to accomplish parameter space analysis for the pre-SSI algorithm, according to some embodiments;

FIG. 14 is a flowchart diagram illustrating the steps involved to accomplish parameter space analysis for the post-SSI algorithm, according to some embodiments;

FIG. 15 is a flowchart illustrating the steps involved in handling the parameterization change, according to some embodiments;

FIG. 16 is a flowchart illustrating the steps involved in performing reparameterization, according to some embodiments;

FIG. 17 is a flowchart illustrating the steps involved in performing reconstruction, according to some embodiments;

FIG. 18 is a flowchart illustrating the optional steps involved in performing postprocessing on the reconstructed watertight surfaces, according to some embodiments;

FIG. 19 is a table identifying C_(PSi) types based on bounding significant point pair types, according to some embodiments;

FIG. 20 is an illustration of how each maximal contiguous group of segments of the same type may be combined to form a cluster (i.e., interval) of the same type, according to some embodiments;

FIGS. 21A-21C are illustrations of isoparametric curve sampling and division based on the model space curve, according to some embodiments;

FIG. 21D is an illustration of the linear reparameterization based on the sub-isocurves shown in FIG. 21C, according to some embodiments;

FIG. 21E is an illustration of a non-linear reparameterization scheme, using the sub-isocurves of FIG. 21C, according to some embodiments;

FIGS. 22A-22C illustrate the structural components of a reparametrized parametric surface, according to some embodiments;

FIGS. 23A-23E further illustrate the structural components of a reparametrized parametric surface, according to some embodiments;

FIGS. 24A-B is an illustration of two spline surfaces S₀ and S₁ that intersect in model space, according to some embodiments;

FIG. 25A illustrates an example of the result of a solid modeling Boolean operation with an SSI operation on spline surfaces S₀ and S₁ shown in FIGS. 24A-B, according to some embodiments;

FIG. 25B is a blowup illustration of area 200 detailing the discontinuous gap-replete fashion of the Boolean operation performed as shown in FIG. 25A, according to some embodiments;

FIGS. 26A-C illustrate the details of the result of the SSI operation shown in FIG. 25A in model space and the various parameter spaces, according to some embodiments;

FIG. 27A is an illustration of blowups of areas 300, 500, and 600 for spline surface S₀, with details, according to some embodiments;

FIG. 27B is an illustration of blowups of areas 300, 700, and 800 for spline surface S₁, with details, according to some embodiments;

FIG. 27C is an illustration of blowups of areas 300 and 400, with details, along with a detail of the knot refined curve C_(MS), according to some embodiments;

FIG. 28 is an illustration of a detail of the knot refined curve C_(MS) with sample points shown, according to some embodiments;

FIGS. 29A-D is an illustration of isocurve sampling of surfaces S₀ and S₁, with both versions of untrimmed and trimmed isocurves at C_(MS), according to some embodiments;

FIGS. 30A-D is an illustration of an example of linear isocurve reparameterization for isocurve sampling of surfaces S₀ and S₁, according to some embodiments;

FIG. 31A is an illustration of isocurve sampling of surface S₀ with its parameter space, according to some embodiments;

FIG. 31B is an illustration of a reconstructed T-spline surface S₀ with its respective parameter space, according to some embodiments;

FIG. 31C is an illustration of a reconstructed B-spline surface {tilde over (S)}₀ with its respective parameter space, according to some embodiments;

FIG. 31D is an illustration of isocurve sampling of surface S₁ with its parameter space, according to some embodiments;

FIG. 31E is an illustration of a reconstructed B-spline surface {tilde over (S)}₁ with its respective parameter space, according to some embodiments;

FIG. 32A is an illustration of a T-spline surface {tilde over (S)} as a union of {tilde over (S)}₀ and {tilde over (S)}₁ with its global parameter space, according to some embodiments;

FIG. 32B is an illustration of a B-spline surface {tilde over (S)} as a union of {tilde over (S)}₀ and {tilde over (S)}₁ with its global parameter space, according to some embodiments;

FIG. 33 is an illustration of all the isocurves evaluated for the internal knots of an arbitrarily defined surface, according to some embodiments;

FIG. 34 is an illustration of an extraction domain of an isocurve, and the relationship between the isocurve's parametric and geometric representations, according to some embodiments;

FIG. 35 is an illustration of isocurves split at the trim curve for the extraction domain depicted in FIG. 34, according to some embodiments;

FIG. 36 is an illustration of the construction of a linear reparametrized set of isocurves using isocurve linear reparameterization, according to some embodiments;

FIG. 37 is an illustration of a method for using a longest isocurve as the model for the final form of the reparametrized knot vector, according to some embodiments;

FIG. 38 is an illustration of global parameter space reconstruction and reparameterization, according to some embodiments;

FIG. 39 is a table indicating segment type classification based on bounding significant point pair types, according to some embodiments;

FIG. 40 is an illustration of C₀ knots and corresponding significant point types that may be produced from downstream global reparameterization and reconstruction operations, according to some embodiments;

FIG. 41 is another illustration of C₀ knots and corresponding significant point types that may be produced from downstream global reparameterization and reconstruction operations, according to some embodiments;

FIG. 42 is an illustration of an example of the embedded extensions for the significant point knot type 0, according to some embodiments;

FIG. 43 is an illustration of an example of the embedded extensions for the significant point knot type 1, according to some embodiments;

FIG. 44 is an illustration of an example of the inserted extraordinary point technique, according to some embodiments; and

FIG. 45 is an illustration of an instance wherein the surface of a sphere is modified from a quadball configuration to that of a standard single patch configuration with degenerate points at the poles, according to some embodiments.

While the features described herein are susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the subject matter as defined by the appended claims.

The term “configured to” is used herein to connote structure by indicating that the units/circuits/components include structure (e.g., circuitry) that performs the task or tasks during operation. As such, the unit/circuit/component can be said to be configured to perform the task even when the specified unit/circuit/component is not currently operational (e.g., is not on). The units/circuits/components used with the “configured to” language include hardware for example, circuits, memory storing program instructions executable to implement the operation, etc. Reciting that a unit/circuit/component is “configured to” perform one or more tasks is expressly intended not to invoke interpretation under 35 U.S.C. § 112(f) for that unit/circuit/component.

DETAILED DESCRIPTION OF THE EMBODIMENTS Acronyms

Various acronyms are used throughout the present application. Definitions of the most prominently used acronyms that may appear throughout the present application are provided below:

BREP: Boundary Representation

NURBS: Non-Uniform Rational B-Spline

SSI: Surface-Surface Intersection

Terminology

The following is a glossary of terms used in this disclosure:

Memory Medium—Any of various types of non-transitory memory devices or storage devices. The term “memory medium” is intended to include an installation medium, e.g., a CD-ROM, floppy disks, or tape device; a computer system memory or random access memory such as DRAM, DDR RAM, SRAM, EDO RAM, Rambus RAM, etc.; a non-volatile memory such as a Flash, magnetic media, e.g., a hard drive, or optical storage; registers, or other similar types of memory elements, etc. The memory medium may include other types of non-transitory memory as well or combinations thereof. In addition, the memory medium may be located in a first computer system in which the programs are executed, or may be located in a second different computer system which connects to the first computer system over a network, such as the Internet. In the latter instance, the second computer system may provide program instructions to the first computer for execution. The term “memory medium” may include two or more memory mediums which may reside in different locations, e.g., in different computer systems that are connected over a network. The memory medium may store program instructions (e.g., embodied as computer programs) that may be executed by one or more processors.

Carrier Medium—a memory medium as described above, as well as a physical transmission medium, such as a bus, network, and/or other physical transmission medium that conveys signals such as electrical, electromagnetic, or digital signals.

Computer System—any of various types of computing or processing systems, including a personal computer system (PC), mainframe computer system, workstation, network appliance, Internet appliance, personal digital assistant (PDA), tablet computer, smart phone, television system, grid computing system, or other device or combinations of devices. In general, the term “computer system” can be broadly defined to encompass any device (or combination of devices) having at least one processor that executes instructions from a memory medium.

Processing Element—refers to various implementations of digital circuitry that perform a function in a computer system. Additionally, processing element may refer to various implementations of analog or mixed-signal (combination of analog and digital) circuitry that perform a function (or functions) in a computer or computer system. Processing elements include, for example, circuits such as an integrated circuit (IC), ASIC (Application Specific Integrated Circuit), portions or circuits of individual processor cores, entire processor cores, individual processors, programmable hardware devices such as a field programmable gate array (FPGA), and/or larger portions of systems that include multiple processors.

Automatically—refers to an action or operation performed by a computer system (e.g., software executed by the computer system) or device (e.g., circuitry, programmable hardware elements, ASICs, etc.), without user input directly specifying or performing the action or operation. Thus the term “automatically” is in contrast to an operation being manually performed or specified by the user, where the user provides input to directly perform the operation. An automatic procedure may be initiated by input provided by the user, but the subsequent actions that are performed “automatically” are not specified by the user, i.e., are not performed “manually”, where the user specifies each action to perform. For example, a user filling out an electronic form by selecting each field and providing input specifying information (e.g., by typing information, selecting check boxes, radio selections, etc.) is filling out the form manually, even though the computer system must update the form in response to the user actions. The form may be automatically filled out by the computer system where the computer system (e.g., software executing on the computer system) analyzes the fields of the form and fills in the form without any user input specifying the answers to the fields. As indicated above, the user may invoke the automatic filling of the form, but is not involved in the actual filling of the form (e.g., the user is not manually specifying answers to fields but rather they are being automatically completed). The present specification provides various examples of operations being automatically performed in response to actions the user has taken.

Configured to—Various components may be described as “configured to” perform a task or tasks. In such contexts, “configured to” is a broad recitation generally meaning “having structure that” performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently performing that task (e.g., a set of electrical conductors may be configured to electrically connect a module to another module, even when the two modules are not connected). In some contexts, “configured to” may be a broad recitation of structure generally meaning “having circuitry that” performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently on. In general, the circuitry that forms the structure corresponding to “configured to” may include hardware circuits.

Various components may be described as performing a task or tasks, for convenience in the description. Such descriptions should be interpreted as including the phrase “configured to.” Reciting a component that is configured to perform one or more tasks is expressly intended not to invoke 35 U.S.C. § 112, paragraph six, interpretation for that component.

DETAILED DESCRIPTION Computer-Aided Design (CAD) Systems

Modern computer-aided design (CAD) systems provide an environment in which users can create and edit curves and surfaces. The curves may include spline curves such as B-spline curves, or more generally, Non-Uniform Rational B-Spline (NURBS) curves. The surfaces may include tensor product spline surfaces such as B-spline surfaces, or more generally, NURBS surfaces, or even more generally, T-spline surfaces. T-Spline surfaces allow the freedom of having T-junctions in the control net of the surface. Tensor product spline surfaces have certain very desirable properties that make them popular as modeling tools, e.g., properties such as: localized domain of influence of each surface control point on surface geometry; controllable extent of continuity along knot lines; C^(∞) smoothness between knot lines; and the ability to represent complex freeform geometry in a discrete manner that is intuitive for the user.

Two significant paradigms within computer-aided design with regards to the geometric modeling of an object are surface modeling and solid modeling.

In surface modeling, an object is represented simply as a set of unconnected surfaces, without maintaining a model of the topological structure corresponding to the relationships between the geometric features of the object. Thus, while the user may design two trimmed surfaces within an object that apparently intersect along a given curve, the CAD system does not explicitly record the topological or geometric relationship between the trimmed surfaces. Thus, in surface modeling, surfaces are operated on as independent entities.

In solid modeling, the specification of an object requires the specification of both topology and geometry, which are captured in a data structure referred to as a boundary representation (Brep, B-rep, b-rep or brep). From the topological point of view, a boundary representation includes, at minimum, faces, edges and vertices, and information regarding their interconnectivity and orientation. (An object includes a set of faces. Each face is bounded by a set of edges. Each edge is bounded by a pair of vertices.) From the geometric point of view, the boundary representation includes surfaces, curves and points, which correspond respectively to the faces, edges and vertices of the topological point of view. Thus, a boundary representation provides a way to store and operate on a collection of surfaces, curves and points as a unified object. In addition to faces, edges and vertices, many solid modelers provide additional objects to this data structure, such as loops, shells, half-edges, etc.

Solid modeling may be used to create a 2-manifold or a non-manifold object. A set X is said to be 2-manifold (without boundary) whenever, for every point xϵX, there exists an open neighborhood in X that contains x and is homeomorphic to an open disk of the 2D Euclidean plane. The sphere S² and the torus T² are common examples of 2-manifolds. A set X is said to be a 2-manifold with boundary whenever, for every point xϵX, there exists an open neighborhood in X that contains x and is homeomorphic to either an open disk of the 2D Euclidean plane or the half open disk given by {(x,y): x²+y²<1, and x≥0}. An example of a non-manifold object is the union of two planes. (This object fails the defining condition for a 2-manifold at each point along the intersection of the two planes.) Another example of a non-manifold object is the union of two spheres that touch each other at a single point of tangency.

Solid modeling may also be used to create a 3-manifold having a 2-manifold as its boundary. Common examples of such a 3-manifold include a solid cube, a solid ball and a solid torus.

The process of designing a model for an object often involves the creation of tensor product spline surfaces in a model space, and the application of Boolean operations to the surfaces (or to 3D “solids” that are defined by a 2-manifold B-rep of surfaces, as per above). The Boolean operations require the calculation of intersections between surfaces in the B-rep. The resulting intersections are represented as trimmed surfaces. A fundamental problem with such Boolean operations is that the trimmed surfaces do not meet in a C⁰ continuous fashion at the geometric intersection of the original surfaces. The process involved in determining the geometric location of intersection between two or more tensor product spline surfaces, as well as the creation of the resulting representational geometric objects, is referred to as “surface-surface intersection” (SSI). Using current SSI technology, each trimmed surface exhibits an independent profile of small scale deviations from the true geometric intersection. For example, a user of a CAD system may create spline surfaces S₀ and S₁ as shown in FIG. 1A, and apply one or more Boolean operations to create the trimmed surfaces S_(0,TRIMMED) and S_(1,TRIMMED) shown in FIG. 1B. In this case, the user has specified the one or more Boolean operations to approximately remove from the domain of evaluation of surface S₀ the portion of S₀ that resides in front of surface S₁, and to approximately remove from the domain of evaluation of surface S₁ the portion of S₁ that resides above surface S₀. FIG. 1C is a blowup of the area 100 at the intersection of the trimmed surfaces S_(0,TRIMMED) and S_(1,TRIMMED), exhibiting some of the gaps at the intersection. Thus, Boolean operations typically violate the intuitive expectation that the trimmed surfaces should have a C⁰ continuous interface along the geometric intersection of S₀ and S₁. This is clearly demonstrated by the fact that the trim operations in surface modeling and Boolean operations in solid modeling do not alter the underlying surface representations. Instead, curves are altered and/or created so as to approximately update the domains of evaluation of the original surfaces.

The gaps are a consequence of a basic mathematical fact: the generic intersection of two low-degree polynomial surfaces is a 3D curve of very high degree. For example, the solution to the intersection of two arbitrary bicubic surfaces is a 3D curve of degree 324. While it is theoretically possible to compute the parameters of such a high degree intersection curve based on the parameters of the two polynomial surfaces, that computation would not be realizable within CAD environments given the computing limitations as well as the practicality of using the resulting high degree curve in practice. Thus, when implementing a Boolean operation, a conventional CAD system may:

(a) numerically compute a set of points SOP_(MS) that reside on (or sufficiently near) the surface-surface intersection in the 3D model space (and are scattered across the surface-surface intersection), and/or, compute a set of intersection-related points SOP_(PS0) in the parameter space of spline surface S₀, and/or, compute a set of intersection-related points SOP_(PS1) in the parameter space of the spline surface S₁ (where the subscript “MS” is meant to suggest model space, and the subscript “PS” is meant to suggest parameter space); (b) generate a curve C_(MS) in the 3D model space, based on the set of points SOP_(MS), where the curve C_(MS) approximates the surface-surface intersection, e.g., by interpolating the set of points SOP_(MS); (c) generate a trim curve C_(PS0) in the 2D parameter space of the spline surface S₀, based on the set of points SOP_(PS0), where the trim curve C_(PS0) approximates the preimage of the surface-surface intersection under the spline map S₀, e.g., by interpolating the set of points SOP_(PS0); (d) generate a trim curve C_(PS1) in the 2D parameter domain of spline surface S₁, based on the set of points SOP_(PS1), where the trim curve C_(PS1) approximates the preimage of the surface-surface intersection under the spline map S₁, e.g., by interpolating the set of points SOP_(PS1).

When the Boolean operation is performed in the context of solid modeling, the CAD system will update the boundary representation. The update of the boundary representation may be performed in a variety of ways, e.g., depending on the data structure format of the boundary representation, as well as the solid modeling algorithm applied, e.g., use of Euler operators or a variety of alternative operators. As an illustrative example in view of FIGS. 1A-1C, the boundary representation may be updated by:

adding a new topological edge E and associating the curve C_(MS) with the new topological edge E;

splitting an existing topological face F₀ corresponding to the spline surface S₀ along a new topological coedge E₀ corresponding to the trim curve C_(PS0), to obtain two subfaces F₀ ^(A) and F₀ ^(B);

associating the topological coedge E₀ with the edge E and with the trim curve C_(PS0);

selecting one of the two subfaces F₀ ^(A) and F₀ ^(B) to be retained and the other to be discarded, e.g., depending on the identity of the Boolean operation (intersection, subtraction, or union), and/or, on user input;

replacing the face F₀ with its selected subface F₀ ^(selected), and associating the spline surface S₀ with the selected subface F₀ ^(selected);

splitting an existing topological face F₁ corresponding to the spline surface S₁ along a new topological coedge E₁ corresponding to the trim curve C_(PS1), to obtain two subfaces F₁ ^(A) and F₁ ^(B);

associating the topological coedge E₁ with the edge E and with the trim curve C_(PS1);

selecting one of the two subfaces F₁ ^(A) and F₁ ^(B) to be retained and the other to be discarded, e.g., depending on the identity of the Boolean operation (intersection, subtraction, or union), and/or, on user input;

replacing the face F₁ with its selected subface F₁ ^(elected) and associating the spline surface S₁ with the selected subface F₁ ^(selected).

Observe that there are three available approximations to the true surface-surface intersection: the trajectory of the model space curve C_(MS); the image S₀(C_(PS0)) of the trim curve C_(PS0) under the spline map S₀; and the image S₁ (C_(PS1)) of the trim curve C_(PS1) under the spline map S₁. Each of those approximations exhibits an independent profile of deviations from the true surface-surface intersection SSI_(TRUE), and disagrees with the other approximations:

C _(MS) ≠SSI _(TRUE);

S ₀(C _(PS0))≠SSI _(TRUE);

S ₁(C _(PS1))≠SSI _(TRUE);

S ₀(C _(PS0))≠C _(MS);

S ₁(C _(PS1))≠C _(MS);

S ₀(C _(PS0))≠S ₁(C _(PS1)).

In the above list, the notation X≠Y may be interpreted as meaning that the Hausdorff distance between set X and set Y is greater than zero. The gaps between the trimmed surfaces may be due especially to the inequality S₀(C_(PS0))≠S₁ (C_(PS1)).

In the environment of surface modeling, there may be a separate model space curve CS_(MSk) for each surface S_(k) participating in the intersection. For example, surface S₀ may be trimmed relative to surface S₁, and surface S₁ may be separately trimmed relative to surface S₀. Each trimming operation may generate a corresponding set of intersection points SOP_(MSk) in model space, and thus, result in a separate model space curve CS_(MSk). Thus, to the above multiplicity of set disagreements, we should add, for each k:

C _(MSk) ≠SSI _(TRUE);

C _(MSk) ≠S ₀(C _(PS0));

C _(MSk) ≠S ₁(C _(PS1))

Therefore, a solid modeling Boolean operation on spline surfaces generally results in at least one model space curve and a set of parametric trim curves per spline surface. Because these curves represent separate approximations to the geometric intersection, gaps and openings are introduced along the geometric intersection. Furthermore, the editable surface features of the tensor product spline surfaces (e.g. control points, degree, weights, etc.) are rendered un-editable and static.

The above described problem is known in computer-aided design as the trim problem (or the SSI problem). The trim problem is considered to be one of the top five problems in the field, and many consider it to be the most important problem. This problem affects all users of the resultant model, including the numerous downstream users and applications of the model. There has been no general solution for this problem to date. The scope and weight of the problem may be summarized with an excerpt from, by one of the industry's leading experts on the topic, Tom Sederberg [Sederberg 2008]:

-   -   “The existence of these gaps in trimmed NURBS models seems         innocuous and easy to address, but in fact it is one of the most         serious impediments to interoperability between CAD, CAM and CAE         systems. Software for analyzing physical properties such as         volume, stress and strain, heat transfer, or lift-to-drag ratio         will not work properly if the model contains unresolved gaps.         Since 3D modeling, manufacturing and analysis software does not         tolerate gaps, humans often need to intervene to close the gaps.         This painstaking process has been reported to require several         days for a large 3D model such as an airplane and was once         estimated to cost the US automotive industry over $600 million         annually in lost productivity. At a workshop of academic         researchers and CAD industry leaders, the existence of gaps in         trimmed-NURBS models was singled out as the single most pressing         unresolved problem in the field of CAD.”

Furthermore, the difficulty and long persistence of the trim problem is evidenced by the following assertions made by Christoph M. Hoffmann, one the leading experts in the field of solid modeling, excerpted from [Hoffman 1989]:

-   -   “The difficulty of evaluating and representing the intersection         of parametric surface patches has hindered the development of         solid modelers that incorporate parametric surfaces. Roughly         speaking, the topology of a surface patch becomes quite         complicated when Boolean operations are performed. Finding a         convenient representation for these topologies continues to be a         major challenge.”

Currently there is no direct solution to the trim problem. Instead, most industries have developed alternative surface descriptions that attempt to avoid the issues with SSI. The alternative surface descriptions approximate the originally supplied surface splines. Examples of alternative surface descriptions include subdivision surfaces and faceted remeshing with geometric healing. As another example, T-splines might be constructed to represent a multi-patch NURBS surface geometry, given an appropriate set of conditions. Unfortunately, these alternative surface descriptions introduce additional approximations, introduce many other problems that did not previously exist in the original model, and either do not allow for Boolean operations or provide for Boolean operations in a manner that is unreliable and very difficult to use.

As noted above, without correction, the gaps between the trimmed surfaces of object models would severely compromise downstream processes such as engineering analysis, manufacturing, graphics rendering and animation. For example, in a thermal analysis, the computed flow of heat between the trimmed surfaces would be disrupted by such gaps. In the context of 3D printing, the gaps would translate into locations where material would be not deposited, and the structural integrity of the resulting printed object would be undermined. In graphics, the rendered image would contain the appearance of cracks at the gaps. In animation, the objects and characters would separate when dynamically actuated.

As further noted above, the correction of such gaps between trimmed surfaces unfortunately involves the exertion of intensive human labor. For example, prior to engineering analysis, the boundary representation may be converted into a continuous mesh of triangles (or polygons). Tremendous human labor is typically required to heal the gaps by injecting a myriad of small polygons to fill the area left between surfaces. The gaps may be so numerous and of such small scale that it may be difficult to discover and correct all the gaps in a single pass of human gap correction. Repeated cycles of analysis, machining, rendering, animation, depending on process of interest, may be dedicated to nothing more than identifying locations of gaps not caught by previous iterations of human gap correction, substantially decreasing the time and budget available for meaningful analyses that would inform optimizations of the model design.

Existing solutions for preparing CAD models for analysis include healing software (CADdoctor, Acc-u-Trans CAD, 3D Evolution, etc.) and meshing software (Altair Hypermesh, CFX-Mesh, Ennova). These solutions fail to mitigate the workflow burden since they do not address the fundamental model differences between disciplines. While they claim to expedite the steps in the current CAD-CAE-CAM pipeline, the processes remain semi-automated and require substantial amounts of users' time and skill.

Therefore, there is a vital need for mechanisms capable of performing Boolean operations on surfaces (or solid objects) in a boundary representation so that the resulting surfaces meet in a C⁰ continuous fashion at their geometric intersections. Furthermore, there exists a vital need for mechanisms capable of reconstructing the surfaces in a previously-created boundary representation so that the reconstructed surfaces meet at their geometric intersections in a C⁰ continuous (i.e., conformal) fashion. These mechanisms may enable the benefits of technologies such as isogeometric analysis (IGA) to be more fully and systematically realized.

FIG. 2: Computer System

FIG. 2 illustrates one embodiment of a computer system 900 that may be used to perform any of the method embodiments described herein, or, any combination of the method embodiments described herein, or any subset of any of the method embodiments described herein, or, any combination of such subsets.

In some embodiments, a hardware device (e.g., an integrated circuit, or a system of integrated circuits, or a programmable hardware element, or a system of programmable hardware elements, or a processor, or a system of interconnected processors or processor cores) may be configured based on FIG. 2 or portions thereof. Any hardware device according to FIG. 2 may also include memory as well as interface circuitry (enabling external processing agents to interface with the hardware device).

Computer system 900 may include a processing unit 910, a system memory 912, a set 915 of one or more storage devices, a communication bus 920, a set 925 of input devices, and a display system 930.

System memory 912 may include a set of semiconductor devices such as RAM devices (and perhaps also a set of ROM devices).

Storage devices 915 may include any of various storage devices such as one or more memory media and/or memory access devices. For example, storage devices 915 may include devices such as a CD/DVD-ROM drive, a hard disk, a magnetic disk drive, a magnetic tape drive, semiconductor-based memory, etc.

Processing unit 910 is configured to read and execute program instructions, e.g., program instructions stored in system memory 912 and/or on one or more of the storage devices 915. Processing unit 910 may couple to system memory 912 through communication bus 920 (or through a system of interconnected busses, or through a computer network). The program instructions configure the computer system 900 to implement a method, e.g., any of the method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or any combination of such subsets.

Processing unit 910 may include one or more processors (e.g., microprocessors).

One or more users may supply input to the computer system 900 through the input devices 925. Input devices 925 may include devices such as a keyboard, a mouse, a touch-sensitive pad, a touch-sensitive screen, a drawing pad, a track ball, a light pen, a data glove, eye orientation and/or head orientation sensors, a microphone (or set of microphones), an accelerometer (or set of accelerometers), or any combination thereof.

The display system 930 may include any of a wide variety of display devices representing any of a wide variety of display technologies. For example, the display system may be a computer monitor, a head-mounted display, a projector system, a volumetric display, or a combination thereof. In some embodiments, the display system may include a plurality of display devices. In one embodiment, the display system may include a printer and/or a plotter.

In some embodiments, the computer system 900 may include other devices, e.g., devices such as one or more graphics devices (e.g., graphics accelerators), one or more speakers, a sound card, a video camera and a video card, a data acquisition system.

In some embodiments, computer system 900 may include one or more communication devices 935, e.g., a network interface card for interfacing with a computer network (e.g., the Internet). As another example, the communication device 935 may include one or more specialized interfaces for communication via any of a variety of established communication standards or protocols or physical transmission media.

The computer system 900 may be configured with a software infrastructure including an operating system, and perhaps also, one or more graphics APIs (such as OpenGL®, Direct3D, Java 3D™).

Any of the various embodiments described herein may be realized in any of various forms, e.g., as a computer-implemented method, as a computer-readable memory medium, as a computer system, etc. A system may be realized by one or more custom-designed hardware devices such as ASICs, by one or more programmable hardware elements such as FPGAs, by one or more processors executing stored program instructions, or by any combination of the foregoing.

In some embodiments, a non-transitory computer-readable memory medium may be configured so that it stores program instructions and/or data, where the program instructions, if executed by a computer system, cause the computer system to perform a method, e.g., any of the method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets.

In some embodiments, a computer system may be configured to include a processor (or a set of processors) and a memory medium, where the memory medium stores program instructions, where the processor is configured to read and execute the program instructions from the memory medium, where the program instructions are executable to implement any of the various method embodiments described herein (or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets). The computer system may be realized in any of various forms. For example, the computer system may be a personal computer (in any of its various realizations), a workstation, a computer on a card, an application-specific computer in a box, a server computer, a client computer, a hand-held device, a mobile device, a wearable computer, a computer embedded in a living organism, etc.

Reconstruction of Spline Surfaces to Integrate Surface-Surface Intersection and Provide Inter-Surface Continuity

Current technologies employ methods that only allow for one-way conversion of a CAD model, while also introducing additional complications, data structures, and approximations. Such methods include healing/translation applications, polytope meshing software (quadrilaterals, triangles, etc.), and conversions to alternative formats such as subdivision surfaces. Embodiments described herein may produce objects in the same format as the original model (i.e., as freeform surfaces, not polygonal approximations), making them available for editing in the user's native CAD software. Current technology used by existing CAD, meshing, and healing/translation software applications does not allow users to repair models that retain editable properties.

CAD may be applicable to an increasing number of industries, and it is expected that the CAD industry will continue to expand significantly in the coming years. This growth is anticipated due to the need for enhanced product visualization and increased design process efficiency.

The CAD industry may generally be categorized into four distinct market segments: design, engineering (analysis), manufacturing, and animation. These four segments are served by vertical CAD products, limiting horizontal interoperability between applications and making it difficult for engineers and designers to use a single model for the purposes of analysis, manufacturing, and graphics. Some companies exist solely to heal models and deliver models to customers with suitable formats that meet the constraints of current CAD representations and limitations throughout the modeling pipeline.

Modern engineering practice may rely on CAD systems to create geometric representations of components and assemblies. CAD geometry may become the input to CAE (FEA, CFD, etc.), CAM (CNC tool path generation, additive manufacturing, etc.), animation and graphics, inspection, etc. In a typical process, a designer may create a model of a component and/or system and then an engineer may use this CAD model as input for converting to an analysis-suitable model for use in a CAE program. The fabricator may also convert the CAD model independently to their requirements so as to analyze manufacturability with a CAM program. FIG. 3 illustrates a user creating a CAD model on a computer, which may then be used for construction of a corresponding vehicle part. The overall design process may require many iterations of redesign as results are fed back to modify the original CAD model, and each cycle may require a model translation in the design-analysis-manufacturing workflow with different, specialized users involved in arriving at a satisfactory final design.

To represent the highly freeform nature of complex objects, non-uniform rational B-spline (NURBS) surfaces are commonly used geometric representations in CAD systems. NURBS surfaces are mappings from basis functions in a UV-parameter space to the 3D model space, where the user sees their intended geometric design. The most natural and common method of building unique, multifaceted models in CAD systems is to perform Boolean operations, or unions, intersections and differences, on the freeform NURBS surfaces to build up the distinct details of each design. Numerical solutions of the surface-surface intersections (SSI) in 3D model space may be necessary as the exact intersections of these surfaces may be of a polynomial degree and complexity too large for CAD systems to realistically handle in software or to be of practical use by the user for further shaping. An auxiliary topological procedure may then be used to create an ancillary data structure, called a boundary representation (B-rep), providing an independent and non-geometric means of maintaining soundness and validity of the solid.

FIG. 4: Conventional Solid Modeling Technique

FIG. 4 is an illustration of a conventional method for performing solid modeling Boolean operation, which shows the approximation of the B-rep surface-surface intersection. In conventional methods, the underlying NURBS surfaces are typically not altered or updated to reflect the true surface intersections. Rather, an intersection curve may be associated with the surfaces where the SSI occurs by use of the B-rep data structure. Because the underlying surface geometry is not updated, these numerical intersection algorithms may introduce gaps and openings in the geometry at the intersections Further, because the B-rep data structure separates topology from the geometry, the resulting non-watertight surfaces may be rendered individually un-editable and static. The graphics card hides from the user the fact that the surface definitions are never updated, rather, simply removed from rendering by the graphics display in the CAD application.

FIG. 5: Conventional B-Rep Solid Model of an Actual Automotive Part

FIG. 5 illustrates a conventional B-rep solid model of an actual automotive part (left and right images are of the exact same model). FIG. 5 shows (left) the graphic display within the CAD application rendered by the graphics card and (right) the 1200+surface definitions contained in the data structure. Note the discontinuity and independent order of the surface geometry.

A significant weakness of the current intersection paradigm is that the underlying NURBS surfaces are not updated. As shown in FIGS. 4 and 5, the CAD user may see the intent of the operation performed, not because the object definition has been updated to represent the joined geometric configuration, but because the graphics card provides a means of not showing the unwanted portions of the trimmed surfaces (generally through a rasterized mesh). Although this process has worked for displaying CAD models while the user completes their design, this strategy does not work for those using the CAD model in CAE, CAM, animation, or inspection. For example, FIG. 6 illustrates how gaps and a lack of continuity in a standard B-rep model may hinder animation of a model. In contrast, a gap free, watertight continuous model may be more effectively animated. Users may desire watertight, gap-free models, making a standard CAD model unusable without further processing. As such, an immense effort may be required to heal the model to make it gap-free. This cleanup process may be costly, time consuming and frustrating for those inheriting the CAD model. Each iteration of the design process described above may require the engineer to repair gaps between the disparate surfaces introduced by Boolean operations and translate the geometric model from the native CAD B-rep representation to a valid analyzable form, typically a polygonal mesh. This operation may require repetitive, semi-automated and/or manual examination at fine levels of detail to identify and repair the flaws. This results in a one-way design process hindering design feedback, optimization, and information integrity. Dr. Thomas Sederberg, a seminal researcher and expert in CAD, described both the subtlety and gravity of the problem:

“The existence of these gaps in trimmed NURBS models seems innocuous and easy to address, but in fact it is one of the most serious impediments to interoperability between CAD, CAM and CAE systems . . . . Since 3D modeling, manufacturing and analysis software does not tolerate gaps, humans often need to intervene to close the gaps. This painstaking process has been reported to require several days for a large 3D model such as an airplane.”

FIG. 7—Modeling for Computational Analysis

Embodiments presented herein may address a core obstacle in the field that costs time and money in lost productivity due to repairing models built on B-rep Boolean operations. FIG. 7 illustrates a conventional faceted quadrilateral mesh model as compared to the watertight B-rep model according to embodiments presented herein. Non-watertight models may limit the abilities of, e.g., CAE, CAM, graphics technologies, additive manufacturing/3D printing, metrology and inspection, as they all may need to heal and rebuild, or mesh, the model to mitigate gaps in the CAD geometry. Entire branches of technology and industry have been set up to deal with this expensive and limiting problem, where differences in B-rep solid definitions across applications may break the models' solid integrity upon file translation. The problem is so pervasive that many companies specializing in healing CAD models have been created to serve the market. Studies have shown that engineers spend a significant portion of their time repairing CAD models, and the amount of time spent repairing CAD models continues to grow as the geometry of models become increasingly complex. This is one of the leading drivers in the development and popularity of isogeometric analysis (IGA) within engineering analysis, a method which seeks to make analysis more CAD friendly. By eliminating gaps in models, embodiments presented herein may remove a substantial roadblock and empower IGA as a commercial technology.

Embodiments herein present a watertightCAD methodology that provides geometric and topological reconstruction designed to enhance current B-rep modeling. In some embodiments, output surfaces are generated with surface edges (isocurves) that meet continuously along the SSI curve, creating a gap-free result. Furthermore, the B-rep data may be updated to furnish information about the geometric objects' parameter spaces, hence integrating the geometry and topology divide. This may allow the surfaces of the solid model to become editable in the CAD application, a huge advantage for modelers needing to make local updates to the geometry. The watertight models may be analysis-suitable within the IGA framework, with no additional healing or meshing required.

In some embodiments, subsequent to outputting a modified, watertightCAD model, the method may proceed by executing an engineering analysis on the modified model to obtain data predicting physical behavior of an object described by the model. In some embodiments, the engineering analysis may comprise an isogeometric analysis. In some embodiments, the method may proceed to direct a process of manufacturing an object described by the modified model. In some embodiments, the method may proceed to generate an image of an object based on the modified model. In some embodiments, e.g., in an animation application, the method may proceed to generate a sequence of animation images based on the modified model and display the sequence of animation images.

FIG. 8-9—Watertight CAD Method and Flowchart

FIG. 8 is a graphical illustration of the watertightCAD method, according to some embodiments. As shown, some embodiments presented herein create new NURBS surfaces, based on the original surface geometry, that integrate the surface-surface intersection information completely within a simple NURBS surface description, thus producing a model that matches the geometric and topological design intent. The surface edges may meet at respective isoparametric boundaries (isocurves). In some embodiments, this solution may solve the trim problem natively, cleanly, and in a way that produces objects in the same form as that of the original, hence it may be a closed-form operation. This may make the technology employable in any current CAD system using standard software kernel functionality. As such, the NURBS surface objects that are output may retain editable properties (control points, weights, knots, etc.), which may be unavailable within the present B-rep solid modeling paradigm. In some embodiments, the retention of editable properties may be achieved without introducing any new data structures on the new NURBS surface. The result may not be a polygonal, mesh or STL approximation. Some embodiments may provide a long-awaited alternative to B-rep solid modeling Boolean operations, and may supply a solid model for the CAD user that may be accurate to model tolerance.

FIG. 9 is a flowchart of the watertightCAD method indicating five steps, including the three core technical steps, required to transform a traditional B-rep solid model into a watertight model, according to some embodiments. In some embodiments, the waterightCAD method may be performed as part of a Boolean operation in a computer-aided design (CAD) software system. These steps are illustrated graphically in FIG. 8, where the basis functions S₁ and S₂ are plotted in the (u,v) parameter space. Notice that initially, the S₁ and S₂ basis functions are each expressed in terms of their own unique parameters, (u₁,v₁) and (u₂,v₂), respectively. The reparameterization step then describes a new global set of basis functions S that are expressed in terms of a new global parameter space (ũ, {tilde over (ν)}).

The watertightCAD algorithm may begin by inputting a CAD file comprising a first and second input surface associated with the solid model. Each of the first and second input surfaces may be described in their own parameter space domains, and may be non-uniform rational basis splines (NURBS). For example, the first input surface may be described within a first parameter space domain and the second input surface may be described in a second parameter space domain. A parameter space domain may be understood to be distinct from a parameter space in that a parameter space domain is constrained by a knot vector. The first and second input surfaces may cross in space along at least one intersection curve, and methods herein may seamlessly connect the input surfaces along their intersection.

Subsequent to the input of a CAD file, and as explained in further detail below, preprocessing may be optionally performed on the CAD file at 100. Subsequently, three core technical steps, 200, 300, and 400 may be performed, which may be described as parameter space analysis, reparameterization, and reconstruction, respectively. 200, 300, and 400 may be vital functions in transforming a standard B-rep solid model to a watertight representation.

Parameter space analysis 200 may comprise identifying and storing a model space trim curve that approximates a geometric intersection of the first and second input surfaces. Characteristic points may be identified along the model space trim curve, to aid in the reparameterization process.

Reparameterization 300 may comprise reparametrizing the first and second parameter space domains into a single global parameter space domain. In other words, the first and second parameter space domains may be reparametrized into a common third parameter space domain. The third parameter space domain may be determined based on the model space trim curve. In other words, the third parameter space domain may be determined based on the geometric intersection of the first and second input surfaces. In some embodiments, reparametrizing may comprise constructing a series of isocurves based on the identified characteristic points of the model space trim curve. The constructed series of isocurves may be comprised within the third parameter space domain.

Reconstruction 400 may comprise constructing first and second output surfaces that are each described in the third parameter space domain. The first and second output surfaces may be constructed such that at least a portion of each of their boundaries coincides with the model space trim curve. In some embodiments, a portion of the boundary of each of the first and second output surfaces may be constructed to coincide with the constructed series of isocurves comprised within the third parameter space domain. The concatenation of the first and second output surfaces may therefore be watertight along this portion of each of their boundaries.

These steps may be required for either the pre-SSI or post-SSI algorithms, albeit the specific parameter space processing sub-steps may vary based on the particular algorithm. At 500, postprocessing may be optionally performed on the first and second output surfaces. Finally, the first and second output surfaces may be output as modified surfaces associated with the solid model.

FIG. 10: Pre-SSI and Post-SSI Algorithms

FIG. 10 is a flow chart illustrating the algorithmic steps involved in the watertightCAD methodology that may be performed depending on whether pre-SSI or post-SSI input data is received. The pre-SSI algorithm may assume the user has two B-reps they wish to perform a solid modeling Boolean operation on. The post-SSI algorithm may assume the user has a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. In other words, pre-SSI input data may comprise two B-reps upon which it is desired to perform solid modeling Boolean operations. Post-SSI input data may comprise a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. For both the pre-SSI and post-SSI algorithms, geometric input data may be stored describing a first and second input surface associated with the model under consideration. Each of the first and second input surfaces may be described in a first and second respective parameter space domain.

The input for the Post-SSI algorithm may be a boundary representation created using one or more (typically many) solid modeling Boolean operations on spline surfaces. (The spline surfaces may include tensor product spline surfaces such as NURBS surfaces and/or T-Spline surfaces.) In one embodiment, the boundary representation may be provided in a computer-aided design (CAD) file. In another embodiment, the Post-SSI algorithm may operate as part of the kernel of a CAD system, in which case the boundary representation may have a specialized internal format.

There are many different ways to realize a boundary representation in terms of a data structure. (Different CAD packages have different data structure realizations.) Furthermore, there are many different file formats used to represent the brep data. For execution of the Post-SSI algorithm, the data structure type and/or data file format of the CAD file need not be of any specific standard, whether a product based type or interoperability standard. Some examples of data file formats include, but are not limited to, ACIS (*.sat, *.sab), STEP (*.stp, *.step), Rhino (*.3dm, *.3dmbak), etc.

The Post-SSI algorithm may operate on a boundary representation of an object, and not on alternative representations such as polytope meshes (structured or unstructured), constructive solid geometry (CSG) data, and implicit geometric representations.

Preprocessing

In some embodiments, to enable processing, necessary information may be parsed from the data structure in an optional preprocessing step 100. For the pre-SSI algorithm, at 110, this specifically consists of identifying the intended surfaces involved in the intersection, along with associated topological elements. For the post-SSI algorithm, at 120, this specifically consists of parsing the topological data structure to determine the surfaces involved in the intersection, the trim curves stored in the B-rep, and the topological elements indicating these relationships. For example, for both pre-SSI and post-SSI algorithms, preprocessing may comprise storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve approximates a geometric intersection of the first and second input surfaces. In some embodiments, the model space trim curve may comprise sequential segments of isocurves in a common third parameter space domain, as discussed in greater detail below.

FIG. 11: Preprocessing for the Pre-SSI Algorithm

For embodiments wherein pre-SSI input data is received, the preprocessing algorithm may begin at 111, wherein the specific topological elements (faces, loops, edges, vertices) involved in the SSI operation may be identified.

Subsequently, at 112, faces may be identified from the identified surfaces of 111.

At 113, the faces may be used to identify associated topological elements. For example, loops, edges and vertices may be identified from the identified faces of 112.

FIG. 12: Preprocessing for the Post-SSI Algorithm

For embodiments wherein post-SSI input data is received, the preprocessing algorithm may proceed differently from when pre-SSI input data is received. In particular, at 121, specific topological elements (faces, loops, edges, vertices) involved in the SSI operation may be determined.

At 122, from the determined edges of 121, 3D model space trim curves may be identified. For example, the model space trim curves may approximate geometric intersections of the input surfaces. In some embodiments, the model space trim curve may be constructed based on the characteristic points of the first and second input surfaces and the geometric intersection of the first and second input surfaces. In some embodiments, the model space trim curve may be received as an output from a previous CAD Boolean operation on the first and second input surfaces.

At 123, 2D trim curves in respective UV parameter spaces may be identified from the determined loops of 121.

Finally, at 124, NURBS surfaces may be identified from the determined faces of 121.

In some embodiments, these tasks may be accomplished with suitable calls to a native CAD application programming interface (API). The preprocessing steps described in 110 and 120 are standard tasks in any CAD solid modeler, and are well known in the art.

Parameter Space Analysis

After performing a proper topological analysis of the B-rep model to isolate critical elements, in some embodiments, a subsequent step may be to analyze the trim curve data in the individual parameter spaces in order to provide the requisite data to define a global UV parameter space. This data may comprise classified trim curve segments reorganized in the B-rep structure. As illustrated in FIG. 10, the parameter space analysis step 200 may proceed differently depending on whether pre-SSI (210) or post-SSI (220) input data is received.

The output from either algorithm's (pre-SSI or post-SSSI) parameter space analysis is a model space trim curve yielding the knot vector and isocurve sampling points that are desirable for completion of the watertight surface operations. In some embodiments, the model space trim curve comprises sequential segments of isocurves in a global parameter space domain.

FIG. 13—Parameter Space Analysis for the Pre-SSI Algorithm

FIG. 13 is a flowchart diagram illustrating the steps involved to accomplish parameter space analysis for the pre-SSI algorithm, according to some embodiments. In the pre-SSI algorithm, the intersecting surfaces may be used to create a model space trim curve optimized for the operations in the downstream reparameterization and reconstruction steps.

At 211, for each surface, all of the characteristic points may be solved for as topologically ordered point sets in respective parameter spaces. Characteristic points may be common in any robust SSI operation, identified by root solving.

At 212, from the ordered point sets, classification and clustering operations may be performed. Segments may be classified based on the characteristic point type (derivatives with respect to dominant parametric direction) and may indicate whether they produce u-type or v-type edges, by use of a lookup table.

At 213, surface knots may be added to ordered point sets through cluster information.

At 214, point sets may be projected and exchanged for each surface. The topology may be checked and corrected. Model space subsets may be formed.

At 215, an intersection curve may be created by various conventional methods. The intersection curve may interpolate the model space point set, forming a knot vector.

At 216, the model space points may be evaluated from curve Greville abscissae. Respective surface intersection points may be determined through various conventional methods.

FIG. 14: Parameter Space Analysis for the Post-SSI Algorithm

FIG. 14 is a flowchart diagram illustrating the steps involved to accomplish parameter space analysis for the post-SSI algorithm, according to some embodiments. In the post-SSI algorithm, the parameter space and model space trim curves supplied by the conventional SSI operation performed prior to input may be used for the operations in the downstream reparameterization and reconstruction steps.

In the post-SSI algorithm, since the Boolean operation has been conventionally performed already by a CAD application, there exists trim curve data to analyze, both in parameter space as well as in model space. Because of this, a first step may be to analyze the trim curve data in the individual parameter spaces in order to provide the requisite data to define a global UV parameter space. This data may comprise classified trim curve segments reorganized in the B-rep structure.

Parameter space analysis for the post-SSI algorithm begins at 221, wherein specific points on trim curves and associated data are determined. In some embodiments, this may comprise developing a root solver for computing trim curve derivatives. The determined specific points on trim curves may include: characteristic points (or significant points) based on derivative values (tangent vectors), surface knot locations, as well as trim curve knot data. Characteristic points may be common in any robust SSI operation, solved by root solving, while the surface and curve knots may be particular to embodiments described herein.

At 222, segments between characteristic points found in 2.1 may be identified and classified. Segments of the model space trim curve may be identified and classified based on the characteristic points, wherein each identified segment comprises an isocurve in the third parameter space domain, wherein said classification determines a parameter in the third parameter space domain that has a constant value along the isocurve. Segments may be classified based on the characteristic point type (derivatives with respect to dominant parametric direction) and may indicate whether they produce u-type or v-type edges. This functionality may be missing from current CAD applications and may be enabled by use of a lookup table.

Finally, at 223, the B-rep data structure may be updated by conventional topological operations (i.e. Euler operators) allowing subdivision of the segmented trim curves.

Reparameterization

During reparameterization, a new parameter space may be defined to reproduce the original surface geometry within model tolerance while maintaining surface continuity and explicitly integrating SSI data. For example, reparameterization may comprise reparametrizing the first and second parametric into a common third parameter space domain based on the previously determined model space trim curve. This procedure may utilize isocurves sampled in sets from the source surfaces (e.g., the first and second input surfaces) which may be then collectively reparametrized to define a newly constructed (third) global parameter space. In other words, reparametrizing may comprise, for each of a set of isocurve sampling points, sampling an isocurve, wherein each sampled isocurve intersects another perpendicular isocurve at the isocurve sampling point. Additionally, the reparameterization function may have many degrees of freedom that may require definition as a nonlinear mapping between parameter spaces in order to produce an operation that is smooth, continuous, and of minimal complexity.

In some cases, transforming UV parameter space coordinates from a group of independent, trimmed surfaces in a B-rep solid model to a single watertight definition may be challenging because there is no unique solution. Based on surface reconstruction methodology leveraging isocurve data, embodiments presented herein employ reparameterization operations established for individual NURBS curves. A more direct mathematical formulation may use a composite function. The resulting isocurves of this composition may become complicated, arbitrary functions of an unpredictable form that cannot be guaranteed to be representable in a CAD system. Alternatively, reparameterization functions may be utilized that are themselves B-splines, along with a procedure to create a NURBS representation of the resulting isocurves. Such concepts are well documented and procedures for creating these representations may be found in references on NURBS. Embodiments presented herein introduce a novel approach by applying these methods to repair gaps in trim curves.

It may be difficult in some cases to develop functions for reparametrizing full sets of isocurves. For example, reparameterization may not be simply extending current theory by means of a batch process, but involve techniques replete with their own unique constraints and degrees of freedom. An essential feature of embodiments presented herein is to correctly map the parameter space of a family of isocurves to a new domain of a specified knot vector while maintaining the exact geometric trace in model space. The parametric conversion may require defining reparameterization functions whose construction are central to the success of the watertight method.

In prior implementations, the possibility of error and incompatibility is introduced if independent isocurve reparameterizations are not monitored and regulated. Embodiments presented herein may constrain the reparameterization task to maintain a valid output. In some embodiments, a reparameterization surface function may be defined as a bivariate tensor product spline surface, fully mapping the source to target UV parameter spaces. This function may be strictly monotonic in the direction of the reparametrized variable, may interpolate the points defined from pairs of source and target knot values for each isocurve, and the reparameterization function itself may be built from the knot vectors of the target parameterization (e.g., the Greville criteria). Although the reparameterization function may be linear or piecewise linear, a nonlinear description may be desirable. However, a nonlinear description may be harder to construct as there are an infinite number of candidates that fulfill the above criteria. The method of construction determined may ultimately contribute attributes to the corresponding isocurves used for reconstruction, so correctly defining the reparameterization function may be very important.

Tasks associated with reparameterization may be comprised of two main steps: parameterization change handling, followed by isocurve reparameterization and global parameter space reconstruction. Prior to the actual reparameterization step, it may be desirable to handle changes in parameterization. Specifying extraordinary points (E.P.s) in unstructured meshes may be identified in this manner. E.P.s may be introduced by use of an intermediate trim curve for the purposes of modifying the local topology.

FIG. 15: Parameterization Change Handling

FIG. 15 is a flowchart illustrating the steps involved in handling the parameterization change, according to some embodiments. At 311, parameterization change handling is performed based on specification of the handling type: E.P., embedded extensions, or C⁰. Parameterization change handling may also be performed based on a null algorithm, which may be a pass-through algorithm that does nothing.

At 312, for E.P. or embedded extension handling types, intermediate trim curves are constructed. The intermediate trim curves may be comprised within at least one of the first and second input surfaces. The extraordinary point insertion algorithm may introduce an intermediate trim curve that intersects the model space trim curve and is orthogonal to the model space trim curve at the intersection. The embedded extensions algorithm may perform tangent extension without introducing extraordinary points.

At 313, the rest of the reparameterization and reconstruction operations (e.g., steps 320 and 400) may be performed based on these intermediate trim curves before they are performed for the trim curve originally identified for processing in the parameter space analysis, but for this updated surface based on the parameterization change handling.

FIG. 16: Reparameterization:

FIG. 16 is a flowchart illustrating the steps involved in performing reparameterization, according to some embodiments. Subsequent to the parameterization change handling steps of 311-313, the following may be performed.

At 321, isoparametric curves (isocurves) may be sampled from the original surfaces at locations determined in 200.

At 322, a bivariate reparameterization function may be specified by user input or by pre-calculated criteria, defining a global parameter space. The bivariate reparameterization function may be based on an interpolation strategy, and the interpolation points may be defined from pairs of isocurve source and target knot values. In some embodiments, it optimization characteristics of the tensor-product reparameterization surface function (curvature, continuity, knot complexity) may be determined. These criteria may serve as critical metrics in evaluating performance.

At 323, each isocurve may be reparametrized based on the reparameterization function. In some embodiments, the reparameterization function may be a basis spline function that maps each of the first and second parameter spaces into the common third parameter space.

In some cases, an additional important aspect of the reparameterization step may be the need to handle extraordinary points (E.P.s) in unstructured meshes. Embodiments herein may allow for specifying E.P.s by use of an intermediate trim curve for the purposes of modifying the local topology. This has traditionally been shown to be effective in standard tests (e.g., hole-in-the-plate), yet may require intervention to specify the bounds of the operation for general modeling. In order to avoid such manual specifications, we may identify a natural and repeatable transition in parameterization for arbitrary models that can easily be implemented in our programming framework. To accomplish this objective the following tasks may be completed:

3.2.1. Inspect and record E.P. reparameterization bounds through focused analysis of models furnished by industrial partners. The bounds may present themselves at characteristic points identified near E.P.s that spread to the surrounding parameterization.

3.2.2. Formalize and validate the measured criteria from task 3.2.1 with respect to theory. The criteria may be aligned to the current research field on E.P.s.

3.2.3 Improve theory from 3.2.2 by application to models in 3.2.1.

FIG. 17: Reconstruction

FIG. 17 is a flowchart illustrating the steps involved in performing reconstruction, according to some embodiments. With a global parameter space created from the input B-rep SSI information, the model space geometry may be reconstructed to correctly represent the intersection requirements. In implementations of the present invention, it may be desirable for the surfaces to meet in a C⁰ continuous fashion across their geometric intersections, and for the original surface geometry to be maintained to model tolerance. In other words, the algorithm may proceed to construct first and second output surfaces, wherein the first and second output surfaces are described in the third parameter space domain. The first and second output surface may be constructed such that at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve. In some embodiments, the first and second output surfaces may be constructed within a predetermined error tolerances of the first and second input surfaces, respectively. In some embodiments, the predetermined error tolerance may be user-defined.

In some embodiments, reconstruction may begin at 401, wherein reparametrized isocurves may be grouped based on target reconstruction surface domains.

At 402, control points of the individual isocurves may be arranged in matrices.

At 403, reconstructed surface control points may be solved for based on isocurve control points.

At 404, from the solved control points of the reconstructed surfaces, along with the global parameter space, reconstructed surfaces may be defined.

FIG. 18: Postprocessing

FIG. 18 is a flowchart illustrating the optional steps involved in performing postprocessing on the reconstructed watertight surfaces, according to some embodiments. In some embodiments, to complete the closed operation, it may be desirable for the model components to be reorganized within the standard B-rep data structure and returned to the native CAD application. In some embodiments, repopulating the B-rep data structure may proceed with the following tasks:

At 501, the topological elements (faces, loops, edges, vertices) may be updated.

At 502, the disparate surfaces may be replaced with reconstructed watertight surfaces.

At 503, the “trim curves” may be updated with isocurves and knots for B-rep agreement.

Subsequent to completion of the postprocessing steps, a watertightCAD file comprising the postprocessed output surfaces may be output from the algorithm.

Further Technical Algorithmic Detail

The following paragraphs present a more detailed and technical explanation of the steps involved in the watertightCAD method, according to some embodiments.

Traverse the Topology of the Boundary Representation to Access SSI-Related Data

Detailed herein is further exposition on the preprocessing method for the Post-SSI algorithm, 120, according to some embodiments.

In some embodiments, the Post-SSI algorithm may traverse the boundary representation to identify a set of topological entities associated with a surface-surface intersection, e.g., a pair of topological faces F₀ and F₁ that share a topological edge E. This identification may be performed in any of a variety of ways, e.g., depending on the data structure format of the boundary representation. The face F₀ is associated with a spline surface S₀ and one or more parameter space trim curves whose images reside in the 2D parametric domain of the spline surface S₀. (The boundary representation may store spline curves and spline surfaces in terms of geometry data sets. For example, a geometry data set for a curve may include its knot vector and its curve control points. The geometry data set for a surface may include its knot vector pair and its surface control points.) One of those parameter space trim curves, which we denote as C_(PS0), is associated with the edge E. Likewise, the face F₁ is associated with a spline surface S₁ and one or more parameter space trim curves whose images reside in the 2D parametric domain of the spline surface S₁. One of those parameter space trim curves, which we denote as C_(PS1), is associated with the edge E. Furthermore, the edge E is associated with a curve C_(MS) in model space.

As described below, the Post-SSI algorithm may operate on the surface splines S₀ and S₁ to respectively create a first set of one or more output surface patches and a second set of one or more output surface patches, where a boundary of a union of images of the first set of output surface patches and a boundary of a union of the images of the second set of output surface patches meet in a C⁰ continuous fashion along the model space curve C_(MS). Thus, the gaps in the original surface-surface intersection will have been repaired.

The spline surfaces {S_(i)} may be tensor product spline surfaces of degree combination p_(i)×q_(i), where p_(i) and q_(i) are integers greater than zero, where p_(i) denotes the degree in one parametric dimension, and q_(i) denotes the degree in the other parametric dimension. For example, in some embodiments p_(i)=q_(i)=3, i.e., the surfaces are bicubic surfaces. In other embodiments, the surfaces may be biquartic or biquintic surfaces, or have mixed degrees. In some embodiments, the surfaces may have different degree combinations. For example, one surface might be bicubic while another surface is biquartic.

The object modeled by the boundary representation may include a plurality of topological face pairs that correspond to surface-surface intersections. For each such face pair, the steps detailed below may be performed to construct sets of output surface patches that meet in a C⁰ continuous fashion along their respective model space SSI curves. Thus, the Post-SSI algorithm may reconstruct all the original spline surfaces along the surface-surface intersections in the model, making the model ready for any of various applications, without the need for human intervention for gap repair.

The spline surface S_(i)=S_(i)(s^(i), t^(i)) is defined on a domain

Ω=[S _(min) ^(i) ,S _(max) ^(i) ]×[t _(min) ^(i) ,t _(max) ^(i)]⊂

²

and has knot vectors

^(i) and

^(i).

Segmentation and Classification

The following paragraphs provide additional detail regarding the parameter space analysis steps for the Post-SSI algorithm, 220.

For each spline surface S_(i) involved in the surface-surface intersection, the corresponding parameter space trim curve C_(PSi) (i.e., the trim curve in the 2D parametric domain of the spline surface S_(i)) may be segmented. The segmentation process may partition the parameter space trim curve C_(PSi) into one or more segments so that each segment can easily be classified as being either, “more conveniently viewed as a function of the first surface parameter s^(i)”, or, “more conveniently viewed as a function of the second surface parameter t^(i)”. The segmentation process may be carried out using criteria that identify specific points on the parameter space trim curve C_(PSi). When operating on a neighborhood of a given segment, the surface reconstruction process may replace the segment with either an {tilde over (s)}^(i) isocurve or a {tilde over (t)}^(i) isocurve within a new parametric surface domain, depending on the classification of the segment.

Along with segmenting and classifying the parameter space trim curve C_(PS1), a check is made to identify points on the parameter space trim curve known as extraordinary points (also known as singular points or degenerate points of a tensor product surface patch edge).

Find Significant Points for the Parameter Space Trim Curve Corresponding to Spline Surface

The following paragraphs provide additional detail regarding the parameter space analysis steps for the Post-SSI algorithm, 220.

Given the parameter space SSI curve C_(PSi)(u^(i))=[s^(i)(u^(i)), t^(i)(u^(i))]^(T) corresponding to the spline surface S_(i)(s^(i), t^(i)), identify significant points of a number of different types. (The superscript i in the parameter u^(i) of the parameter space SSI curve C_(PSi) is a reminder that the trajectory of C_(PSi) resides in the domain Ω^(i) of the spline surface S₁.)

In some embodiments, there are four types of significant points, as follows.

Type 0: The significant points of type 0 include the end points and internal kinks and cusps of the curve C_(PSi). In other words, a significant point of type 0 corresponds to a knot of valence greater than or equal to the degree p_(u) ^(i) of the curve C_(PSi)(u^(i)).

Type 1: The significant points of type 1 include the inflection points of the curve C_(PSi), where an inflection point is a point C_(PSi)(u^(i)) such that the slope relative to surface parameter s^(i) is equal to one or minus one:

${{\frac{\left( t^{i} \right)^{\prime}\left( u^{i} \right)}{\left( s^{i} \right)^{\prime}\left( u^{i} \right)}} = {{{{dt}^{i}\text{/}{ds}^{i}}} = 1}},$

where (t^(i))′(u^(i)) denotes differentiation of t^(i) with respect to u^(i), and similarly, (s^(i))′(u^(i)) denotes differentiation of s^(i) with respect to u^(i).

Type 2: The significant points of type 2 include the stationary critical points of the curve C_(PSi), i.e., points C_(PSi)(u^(i)) such that

$\frac{{dt}^{i}}{{du}^{i}} = 0.$

In other words, a stationary critical point is a point where the curve C_(PSi) has zero slope, when viewed as a function of s^(i).

Type 3: The significant points of type 3 include the non-stationary critical points of the curve C_(PSi), i.e., points C_(PSi)(u^(i)) such that

$\frac{{ds}^{i}}{{du}^{i}} = 0.$

In other words, a non-stationary critical point is a point where the curve C_(PSi) has infinite slope, when viewed as a function of s^(i).

Special Cases in Significant Point Analysis

It should be observed that it is possible for the parameter space trim curve C_(PSi) to have one or more line segments as part of its trajectory. If the slope of such a line segment is a member of the set {0, 1, −1, ∞}, that line segment would result in an infinite number of significant points. Thus, such segments of special slope should be removed from the curve C_(PSi) prior to the identification of significant points. Methods to accomplish this identification and removal of line segments from the trajectory of a curve are well known in the art of curve processing.

Classification of Sections of Parameter Space Trim Curve C_(PSi)

The following paragraphs provide additional detail on 222 from FIG. 14, wherein segments between characteristic points found in 221 are identified and classified.

Each significant point of the curve C_(PSi)=C_(PSi)(u^(i)) corresponds to a respective value of the curve parameter u^(i). The significant points of the curve C_(PSi) may be ordered according to the curve parameter u^(i). This ordering partitions the curve C_(PSi) into segments. We use the index α to identify the significant points according to this ordering. Each segment is bounded in u^(i) by a successive pair (u_(α) ^(i), u_(α+1) ^(i)) of the significant points.

Each segment may be classified as being of type s^(i) or type t^(i) based on the types of its endpoints, e.g., as shown in FIG. 19. If the segment is bounded by significant points C_(PSi)(u_(α) ^(i)) and C_(PSi) (u_(α+1) ^(i)), there are sixteen possible cases for the ordered pair

(Type C _(PSi)(u _(α) ^(i)),Type C _(PSi)(u _(α+1) ^(i))),

where Type C_(PSi)(u_(α) ^(i)) denotes the type of the significant point C_(PSi)(u_(α) ^(i)), and Type C_(PSi) (u_(α+1) ^(i)) denotes the type of the significant point C_(PSi) (u_(α+1) ^(i)). As shown in FIG. 19, the type of the segment is immediately resolved in certain cases of those sixteen cases, which are labeled with s^(i) or t^(i). In other cases, the type of the segment must be checked as follows. Calculate the velocity vector

$\frac{{dC}_{PSi}}{{du}^{i}} = \left\lbrack {\frac{{ds}^{i}}{{du}^{i}},\frac{{dt}^{i}}{{du}^{i}}} \right\rbrack^{T}$

at any point in the interior of the segment (i.e., excluding the end points), and classify the segment based on:

${{Segment}\mspace{14mu} {Type}} = \left\{ {\begin{matrix} {{{type}\mspace{14mu} s^{i}},} & {{{{if}\mspace{20mu} {{{dt}^{i}\text{/}{ds}^{i}}}} \leq 1}\mspace{11mu}} \\ {{{type}\mspace{14mu} t^{i}},} & {{{if}\mspace{20mu} {{{dt}^{i}\text{/}{ds}^{i}}}} > 1} \end{matrix}.} \right.$

Note that the choice of assigning the boundary case |dt^(i)/ds^(i)|=1 to type s^(i) is arbitrary. It could equally well have been assigned to type t^(i).

Furthermore, observe that the segment type classification is invariant under exchange of the endpoint types. For example, the segment type of pair (2,1) is the same as the segment type of pair (1,2). Finally, note that the type pairs (3,2) and (2,3) are not possible if the segmentation has been performed properly. (Each represents a transition between zero slope and infinite slope, which will not occur on a continuous curve segment without an intermediate significant point of type 1.) Thus, an error warning may be issued if either of these type pairs is encountered.

Clustering Segments of Same Type into Intervals

In some embodiments, each maximal contiguous group of segments of the same type may be combined to form a cluster (i.e., interval) of the same type, e.g., as shown in FIG. 20. As another example, given a sequence of six segments having the corresponding sequence of types {s^(i), s^(i), s^(i), t^(i), t^(i), s^(i)}, then the first three segments may be combined to form a cluster of type s^(i); the next two segments may be combined to form a cluster of type t^(i); and the last segment would constitute a singleton cluster of type s^(i).

In some embodiments, certain aspects of the processing to be described below may be performed on the basis of a cluster.

Check for Extraordinary Point Locations at Ends of Loop Component Curves and at Kinks/Cusps of the Parameter Space Curve

As part of the traversal of the boundary representation, the Post-SSI algorithm may identify a loop or loops associated with each face corresponding to spline surface S₁ involved in the intersection. The loop may be directed and include component curve(s), among which is the parameter space curve C_(PSi). Each of the component curve(s) maps from a corresponding 1D domain to the 2D parametric domain of the spline surface S₁. The loop partitions the 2D parametric domain into regions. The orientation of the loop is typically used by the boundary representation as a mechanism to indicate which region(s) is to be retained. The other region(s) is discarded (i.e., counted as not being part of the modeled object).

For each successive pair of component curves (C_(LC(n)), C_(LC(n+1))) in the loop, the point where the two component curves (of the pair) meet may be tested for degeneracy as follows. (The notation LC(n) is meant to suggest the n-th component curve of the loop. When the index value n points to the last component curve of the loop, the index value n+1 is interpreted as pointing to the first component curve of the loop.) Let vector vel_(a) denote the velocity vector of component curve C_(LC(n)) at its end, and let vector vel_(b) denote the velocity vector of component curve C_(Lc(n+1)) at its start. In other words, if we denote the parameter of loop component C_(Lc(n)) as r^(n), C_(LC(n))=C_(LC(n))(r^(n)), then

${{vel}_{a} = {{C_{{LC}{(n)}}^{(1)}\left( r_{end}^{n} \right)} = {\frac{{dC}_{{LC}{(n)}}}{{dr}^{n}}\left( r_{end}^{n} \right)}}},{{vel}_{b} = {{C_{{LC}{({n + 1})}}^{(1)}\left( r_{start}^{n + 1} \right)} = {\frac{{dC}_{{LC}{({n + 1})}}}{{dr}^{n + 1}}{\left( r_{start}^{n + 1} \right).}}}}$

Then the meeting point, C_(Lc(n))(r_(end) ^(n))=C_(LC(n+1))(r_(start) ^(n+1)), is declared to be an extraordinary point if

${135^{{^\circ}} \leq {\cos^{- 1}\left( \frac{{vel}_{a} \cdot {vel}_{b}}{{{vel}_{a}}{{vel}_{b}}} \right)} \leq 225^{{^\circ}}},$

where vel_(a)·vel_(b) denotes the dot product of vectors vel_(a) and vel_(b), and ∥vel_(a)∥ and ∥vel_(b)∥ denote the Euclidean norm of vectors vel_(a) and vel_(b), respectively.

If the loop arises as part of a non-manifold object, the loop orientation may not be consistent with the above test criterion because of the particular definition employed for non-manifold curve orientations. The extraordinary point criterion may be realized in different ways, dependent on the specific implementation, but will not be unrealizable because of these variations.

Test for Extraordinary Points at Kinks/Cusps.

Each kink and cusp along the parameter space curve C_(PSi) may be tested for degeneracy. Let u_(ψ) ^(i) be the parametric location of a kink or cusp in the parameter space curve C_(PSi), and let vector vel_(a) denote the derivative of the parameter space curve C_(PSi) evaluated at location u_(ψ) ^(i)−δ, and let vector vel_(b) denote the derivative of the parameter space curve C_(PSi) at location u_(ψ) ^(i)+δ, where δ is a small positive real number.

${{vel}_{a} = {{C_{PSi}^{(1)}\left( {u_{\psi}^{i} - \delta} \right)} = {\frac{{dC}_{PSi}}{{du}^{i}}\left( {u_{\psi}^{i} - \delta} \right)}}},{{vel}_{b} = {{C_{PSi}^{(1)}\left( {u_{\psi}^{i} + \delta} \right)} = {\frac{{dC}_{PSi}}{{du}^{i}}{\left( {u_{\psi}^{i} + \delta} \right).}}}}$

Then, the point C_(PSi)(u_(ψ) ^(i)) may be identified as an extraordinary point if vel_(a) and vel_(b) satisfy the above stated angular condition.

Refinement Steps and Boundary Isocurve Reparameterization

In the following steps, the Post-SSI algorithm may refine the surfaces {S_(i)} and the model space curve C_(MS), i.e., apply knot refinement to these entities. Each of these entities may be refined based on both intrinsic location information and external location information. When refining an entity based on location information that naturally belongs to that entity, the location information is referred to as intrinsic location information. For example, when refining a surface (or curve) based on its original knot values, the knot values are interpreted as intrinsic location information. As another example, when refining a surface based on the significant points residing within the parametric domain of that surface, the significant points may be interpreted as intrinsic location information. Conversely, when refining an entity based on location information that naturally belongs to another entity, the location information is referred to as external location information. For example, when refining a surface based on the significant points residing in the parametric domain of another surface, the significant points are interpreted as external location information. As another example, when refining the model space curve C_(MS) based on the knots of a surface, the knots are interpreted as external location information.

Furthermore, as part of the geometry reconstruction process to be described below, we may create two output tensor product surface patches such that (a) their boundaries meet continuously along a portion of the model space curve and (b) the union of the images of the two output surface patches approximates a neighborhood in the image of the spline surface S_(i), i.e., a neighborhood that straddles the surface-surface intersection. It would be desirable if isocurves in the first output surface patch could meet the respective isocurves of like parametric value in the second output surface patch, and if the point of meeting were without parametric derivative discontinuity. (The component of the first derivative in the direction tangent to the model space curve is preferably without discontinuity at the point of meeting.) In order to enforce these continuity conditions on the isocurves, isoparametric boundary curves in the complementary parameter direction may be reparametrized to achieve parametric agreement with the model space curve. These isoparametric boundary curves may be reparametrized in a variety of ways, e.g., depending on a number of features desired by the user.

Refinement of Surfaces and Model Space Curve Based on Intrinsic Location Information.

A) Refine the Model Space Curve by Knot Insertion at Knots of Model Space Curve.

The model space curve C_(MS)(ν) may be refined by performing knot insertion at each interior knot in the original knot vector V of model space curve C_(MS)(ν), to raise the multiplicity of the knot to the degree of the model space curve C_(MS). (Hereinafter, unless stated otherwise, when we refer to “refining” a given curve at a given parametric location, we will mean the process of raising the knot multiplicity at that parametric location to the degree of the given curve. Hereinafter, unless stated otherwise, when we refer to “refining” a given surface at a given location in one of the surface's parametric variables, we will mean the process of raising the knot multiplicity at that location to the degree of the surface in that parametric variable.) As is well understood in the art of spline-based modeling, this operation causes the model space curve to be separated into intervals of simplified form (e.g., of Bézier form), without changing the trajectory of the curve in space.

B) Refine Surface S_(i) by Knot Insertion at s^(i) Knots or t^(i) Knots of Surface S_(i), Depending on Classification.

For a given segment (or cluster) of the parameter space curve C_(PSi), the spline surface S_(i) may be refined at certain interior surface knots of the spline surface S_(i), depending on the classification of the segment. Suppose x^(i) ϵ{s^(i), t^(i)} denotes the type of the segment. The surface S_(i) may then be refined at interior x^(i) knots of S_(i) that occur on the closed interval [x^(i)(u_(α) ^(i)), x^(i)(u_(α+1) ^(i))], where u_(α) ^(i) and u_(α+1) ^(i) denote the parameter values of C_(PSi) (u^(i)) that bound the segment. (An x^(i) knot of the surface S_(i) is either an s^(i) knot if the segment is of type s^(i), or a t^(i) knot if the segment is of type t^(i). Furthermore, a y^(i) knot of the surface S_(i) is either a t^(i) knot if the segment is of type s^(i), or an s^(i) knot if the segment is of type t^(i).)

Recall that the parameter space curve C_(PSi)(u^(i)) has coordinate functions s^(i)(u^(i)) and t^(i)(u^(i)) such that

C _(PSi)(u ^(i))=[s ^(i)(u ^(i)),t ^(i)(u ^(i))]^(T).

Thus, if the segment is of type x ^(i) =s ^(i), then

[x ^(i)(u _(α) ^(i)),x ^(i)(u _(α+1) ^(i))]=[s ^(i)(u _(α) ^(i)),s ^(i)(u _(α+1) ^(i))].

Conversely, if the segment is type x ^(i) =t ^(i), then

[x ^(i)(u _(α) ^(i)),x ^(i)(u _(α+1) ^(i))]=[t ^(i)(u _(α) ^(i)),t ^(i)(u _(α+1) ^(i))].

C) Refine the Surface S_(i) based on the Significant Points of Parameter Space Curve C_(PSi).

Given a segment of type x^(i) ϵ{s^(i), t^(i)} from the parameter space curve C_(PSi), the surface S_(i) may be refined at the x^(i)-coordinate locations corresponding to the significant points that bound the segment. In other words, the surface S_(i) may be refined at the locations x^(i) (u_(α) ^(i)) and x^(i)(u_(α+1) ^(i)), where u_(α) ^(i) and u_(α+1) ^(i) denote the parameter values of curve C_(PSi) (u^(i)) that bound the segment.

Refinement of Surfaces and Model Space Curve Based on External Location Information.

A) Refine the Model Space Curve C_(MS) based on Surface Knots of Each Surface S₁.

For a given segment (or cluster) of type x^(i) ϵ{s^(i), t^(i)} from the parameter space curve C_(PSi), the Post-SSI algorithm may refine the model space curve C_(MS)(ν) at locations corresponding to the interior x^(i) knots of surface S₁ that occur on the closed interval [x^(i) (u_(α) ^(i)), x^(i)(u_(α+1) ^(i))]. Furthermore, in some embodiments, any y^(i) knot of the surface S_(i) whose corresponding knot line intersects the given segment (or cluster) of the parameter space curve C_(PSi) may be transferred to the model space curve C_(MS), and used to refine the model space curve C_(MS).

The above described refinements of the model space curve may be performed for each segment of the parameter space curve C_(PSi) and/or for each surface S_(i) involved in the surface-surface intersection.

An x^(i) knot of S_(i) may be mapped to the one-dimensional parametric domain of the model space curve C_(MS) (ν) in a variety of ways. For example, one may map a surface knot s_(ε) ^(i) onto the domain of C_(MS)(ν) via a minimization of distance between the t^(i) isocurve S_(i)(s_(E) ^(i), t^(i)) and the model space curve C_(MS)(ν)). Similarly, one may map a surface knot t_(η) ^(i) onto the domain of C_(MS)(ν) via a minimization of distance between the s^(i) isocurve S_(i)(s^(i), t_(η) ^(i)) and the model space curve C_(MS)(ν)).

B) Refinement of the Model Space Curve based on Significant Points of Each Parameter Space Curve C_(PSi).

For all values of the surface index i, the Post-SSI algorithm may refine the model space curve C_(MS)(ν) at locations {ν₀,} corresponding to the significant points {(s_(α) ^(i), t_(α) ^(i))} of the parameter space curve C_(PSi) (u_(α) ^(i)). There are a variety of ways to determine the locations {ν_(∝)}. For example, in one embodiment, one may map a significant point (s_(α) ^(i), t_(α) ^(i)) of the parameter space curve C_(PSi) onto the 1D parametric domain of C_(MS)(ν) via a minimization of distance between S_(i)(s_(α) ^(i), t_(α) ^(i)) and C_(MS)(ν), and then point inversion.

C) Refine Each Surface S₁ based on the Knots of the Model Space Curve.

Each surface S_(i) may be refined based on the interior knots of the model space curve C_(MS). Given a knot ν_(ψ) of the model space curve C_(MS), the knot ν_(ψ) may be mapped to a point (s_(ψ) ^(i), t_(ψ) ^(i)) in the 2D parametric domain of the surface S_(i). The point (s_(ψ) ^(i), t_(ψ) ^(i)) may then be mapped to a closest segment of the parameter space curve C_(PSi). If the closest segment is of type x^(i)ε{s^(i), t^(i)}, the surface S_(i) may be refined at x_(ω) ^(i). In other words, if the segment type x^(i)=s^(i), the surface S_(i) is refined at s^(i)=s_(ψ) ^(i); conversely, if the segment type x^(i)=t^(i), the surface S_(i) is refined at t^(i)=t_(ψ) ^(i).

As noted above, a knot value ν_(k) of the model space curve C_(MS) may be mapped to a point (s_(ψ) ^(i),t_(ψ) ^(i)) in the 2D parametric domain of the surface S_(i). There are a variety of ways to perform this mapping. In one embodiment, the knot value ν_(ψ) may be mapped by minimizing the distance between C_(MS)(ν_(ψ)) and S_(i)(s^(i),t^(i)), and then point inversion.

D) Refine the Surface S_(i) Based on External Significant Points.

The Post-SSI algorithm may refine the surface S_(i) based on external significant points, i.e., significant points occurring in the 2D parametric domain(s) of the other participating surface(s). For each significant point (s_(α) ^(j),t_(α) ^(j)) of each other participating surface S_(j), j≠i, a corresponding point (s_(MAP) ^(i),t_(MAP) ^(i)) in the (s^(i),t^(i)) domain of the surface S_(i) may be computed. The point (s_(MAP) ^(i),t_(MAP) ^(i)) may be mapped to a closest segment of the parameter space curve C_(PSi). If the closest segment is of type x^(i)ϵ{s^(i),t^(i)}, the surface S_(i) may be refined at x^(i)=x_(MAP) ^(i). In other words, if the segment type x^(i)=s^(i), the surface S_(i) may be refined at s^(i)=s_(MAP) ^(i); conversely, if the segment type x^(i)=t^(i), the surface S_(i) may be refined at t^(i)=t_(MAP) ^(i).

As noted above, a significant point (s_(α) ^(j),t_(α) ^(j)) of a different participating surface S_(i) may be mapped to a point (s_(MAP) ^(i), t_(MAP) ^(i)) in the (s^(i), t^(i)) domain of the surface S_(i). This mapping may be performed in a variety of ways. For example, in one embodiment, if the significant point (s_(α) ^(j), t_(α) ^(j)) has already been transferred to a value ν_(α) in the domain of the model space curve C_(MS), the point (s_(MAP) ^(i), t_(MAP) ^(i)) may be determined by minimizing the distance between the point C_(MS) (ν_(α)) and the surface S_(i) (s t^(i)), and then point inversion. In another embodiment, the significant point (s_(α) ^(j),t_(α) ^(j)) may be mapped more directly to the domain of the surface S_(i) by minimizing the distance between the point S_(i) (s_(α) ^(j), t_(α) ^(j)) and the surface S_(i)(s^(i), t^(i)), and then point inversion. This minimization may be interpreted as a projection from the surface S_(i) to the surface S_(i), or from the 2D parametric domain of surface S_(i) to the 2D parametric domain of surface S_(i).

For the modeling of a 2-manifold, the trim boundaries of at most two trimmed surfaces can meet along any given curve in model space. However, the principles of the present invention are not limited to the context of 2-manifolds. A non-manifold structure may be created where the trim boundaries of more than two surfaces meet along the model space curve. Thus, in some embodiments, it is possible that there are more than two participating surfaces {S_(i)}.

Selection of Domain for Surface Patch Extraction and Isocurve Extraction

The Post-SSI algorithm may organize the construction output geometry relative to surface S_(i) based on successive segments (or clusters) of the parameter space curve C_(PSi). Within each segment (or cluster), the Post-SSI algorithm may organize the construction of output geometry based on successive portions of the segment (or cluster). In particular, for each portion of the segment (or cluster), a corresponding domain for surface patch reconstruction may be selected, herein referred to as the extraction domain. The extraction domain is a subdomain in the 2D parametric domain Ω^(i) of the spline surface S_(i), and is so named because a set of isocurves may be extracted from the restriction of spline surface S_(i) to the extraction domain. The control points of these isocurves may be used to the construct output surface patch(es) corresponding to the portion of the parameter space curve C_(PSi). The boundary of the extraction domain may be referred to as the extraction boundary.

For each surface S_(i), the extraction domains that correspond respectively to the portions of the parameter space curve C_(PSi) may form a partition of the 2D parametric domain of the surface S_(i), i.e., the extraction domains may be non-overlapping regions and their union may be equal to the 2D parametric domain of the surface S_(i). Here the term “non-overlapping” means that the intersection of different extraction domains has zero area. It should be understood, however, that adjacent extraction domains may meet along their boundaries. For example, the right boundary of one extraction domain may be the left boundary of another extraction domain. Given a segment (or cluster) of type x^(i)ϵ{s^(i), t^(i)} from the parameter space curve C_(PSi), the segment (or cluster) may be partitioned into portions based on a set of division points comprising:

-   -   the interior x^(i) knots of surface S_(i);     -   the x^(i) knots of surface S_(i) that have been derived from the         intrinsic and external significant points;     -   the x^(i) knots of surface S_(i) that have been derived from the         knots of the model space curve C_(MS).

The distinct division points may be ordered, and denoted as {x_(β) ^(i)}, where β is an index according to the ordering. (The superscript i is a reminder that the division points are related to surface S_(i). However, we may drop the superscript when there is no need for that reminder.) Each interval of the form [x_(β) ^(i), x_(β+1) ^(i)] defines a corresponding portion CP_(PS1,β) of the segment (or cluster), and gives rise to a corresponding extraction domain Ω_(β) ^(i). The portion CP_(PSi,β) is the restriction of the parameter space curve CP_(PS1) to the interval [x_(β) ^(i), x_(β+1) ^(i)]. (The notation “CP” is meant to suggest “Curve Portion”.) In some embodiments, the union of the extraction domains {Ω_(β) ^(i)} over the index may be equal to the 2D parametric domain Ω^(i) of the surface S_(i): Ω^(i)=U_(β)Ω_(β) ^(i).

Given a portion CP_(PSi,β), we may define a corresponding rectangular extraction domain Ω_(β) ^(i) as follows. The x^(i) extent of the extraction domain Ω_(β) ^(i) may be set equal to the closed interval [x_(β) ^(i), x₊₁]. (Note that we may refer to a portion with the more simple notation CP_(PS1) when the dependency on β is not necessary for the discussion at hand.) The extent of the extraction domain Ω_(β) ^(i) in the complementary parameter y^(i)ϵ{s^(i),t^(i)}, y^(i)≠x^(i), may be selected in various ways. If we denote the y^(i) extent of the extraction domain Ω_(β) ^(i) by the closed interval [y_(LB) ^(i),y_(UB) ^(i)], we have

Ω_(β) ^(i) =[x _(β) ^(i) ,x _(β+1) ^(i) ]×[y _(LB) ^(i) ,y _(UB) ^(i)].

(The notation “LB” and “UB” are meant to be suggestive of “lower bound” and “upper bound”.)

It may be desirable for the extraction domain Ω_(β) ^(i) to bound the selected portion CP_(PSi,β). The value y_(LB) ^(i) may be selected as a value of parameter y^(i) that is strictly less than the minimum value of the parameter y^(i) achieved by the portion CP_(PSi,β), and the value y_(UB) ^(i) may be selected as a value of parameter y^(i) that is strictly greater than the maximum value of the parameter y^(i) achieved by the portion CP_(PSi,β). For example, the value y_(LB) ^(i) may be selected to be a y^(i) knot of the surface S₁ that is strictly less than said minimum, and the value y_(UB) ^(i) may be selected to be a y^(i) knot of the surface S₁ that is strictly greater than said maximum. As another example, the values y_(LB) ^(i) and y_(UB) ^(i) may be selected so that the extraction domain Ω_(β) ^(i) spans the entire y^(i) extent of the domain Ω^(i) of the surface S₁, i.e., y_(LB) ^(i) is set equal to the first y^(i) knot of the surface S_(i) and y_(UB) ^(i) is set equal to the last y^(i) knot of the surface S_(i). As yet another example, the values y_(LB) ^(i) and y_(UB) ^(i) may be selected based on user input.

The Post-SSI algorithm may extract a surface patch S_(i,β) from the spline surface S_(i), i.e., a surface patch corresponding to the extraction domain Ω^(i). (The extraction may involve knot insertion at y_(LB) ^(i) and y_(UB) ^(i), if such insertion has not already been performed.) The image of the surface patch S_(i,β) in model space is coincident with the image of the extraction domain Ω_(β) ^(i) under the spline map S_(i).

In some embodiments, previous steps in the Post-SSI algorithm will have systematically exchanged refinement location information such as knot values and significant points between exchange participants (i.e., the model space curve and surfaces). Thus, the above-described distinct division points {x_(β) ^(i)} along the x^(i) axis of the surface S_(i) will exist in one-to-one correspondence with division points {ν_(β)} in the 1D domain of the model space curve, and in one-to-one correspondence with division points {x_(β) ^(j)} in each other surface S_(j), j≠i. Thus, the x^(i) extent of the extraction domain Ω_(β) ^(i) may correspond not only to the portion CP_(PS1,β) of the parameter space curve CP_(PSi), but also to a portion CP_(MS,β) of the model space curve C_(MS) and a portion CP_(PSj,β) of parameter space curve CP_(PSj) for all j≠i. One may interpret the above described process of exchange and refinement as a common reduction to mutually correlated portions of simplified form.

Reparameterization of Isocurves on the Extraction Boundary.

The following paragraphs present additional detail regarding the reparameterization algorithm, 300.

The Post-SSI algorithm may reparametrize the x^(i) isoparametric boundary curves of the surface patch S_(i,β), e.g., to avoid the shearing of y^(i) isocurves of the reconstructed output patch(s) near the model space curve C_(MS). (In general, x^(i) isoparametric curves may be referred to herein as “transverse isoparametric curves”. Furthermore, y^(i) isoparametric curves may be referred to herein as “longitudinal isoparametric curves”.) In particular, the Post-SSI algorithm may reparametrize the two x^(i) isocurves of the surface patch S_(i,β) corresponding respectively to the boundary values y_(LB) ^(i) and y_(UB) ^(i). We refer to these boundary isocurves as ISO_(y) _(LB) ^(i)(x^(i)) and ISO_(y) _(LB) ^(i)(x^(i)). Each of these boundary isocurves may be reparametrized to achieve parametric agreement with the model space curve. This parametric agreement may be described in terms of a map T_(i) and a projection P_(x) _(i) defined as follows.

The map T_(i) maps from the generic point C_(MS)(ν) on the model space curve C_(MS) to the (s^(i),t^(i)) parametric domain of the surface S_(i). The map T_(i) may be realized, e.g., by distance minimization. The projection P_(x) _(i) maps the point (s^(i),t^(i)) onto the parameter x^(i). If the extraction domain Ω_(β) ^(i) is based on a portion CP_(PSi,β) of type x^(i)=s^(i), then P=s^(i), i.e., the evaluation of P_(x) _(i) on the coordinate pair (s^(i), t^(i)) gives the value s^(i). Conversely, if the extraction domain Ω_(β) ^(i) is based on a portion CP_(PS,β) of type x^(i)=t^(i), then P_(x) _(i) (s^(i),t^(i))=t^(i), i.e., the evaluation of P_(x) _(i) on the coordinate pair (s^(i), t^(i)) gives the value t^(i).

We seek a reparameterization function x^(i)=ƒ_(x) _(i) (ν) that maps from the closed interval I_(β)=[ν_(β),ν_(β+1)] (or any other convenient interval) onto the interval [xβ^(i),x_(β+1) ^(i)] such that, for all ν in I_(β), the following condition is obeyed (or approximated):

P _(x) _(i) T _(i)(C _(MS)(ν))=ƒ_(x) _(i) (ν).

(The expression “x^(i)=ƒ_(x) _(i) (ν)” is interpreted to mean that the evaluation of function ƒ_(x) _(i) on the value ν gives the value x^(i).) The reparameterization function ƒ_(x) _(i) may be linear, piecewise linear or non-linear, and may be computed in any of various ways.

The two boundary isocurves

ISO_(y_(LB)^(i))(x^(i))  and  ISO_(y_(UB)^(i))(x^(i))

are reparametrized using the function ƒ_(x) _(i) to obtain reparametrized isocurves

ISO_(y_(LB)^(i))(v)  and  ISO_(y_(UB)^(i))(v),

respectively. For convenience, in the following steps we may continue to refer to the parameter x^(i), with the understanding that, in at least some embodiments, x^(i) agrees with parameter ν, due to the present reparameterization.

Construction of Output Patch Geometry

The following paragraphs give additional algorithmic detail on the reconstruction steps 400 of the WatertightCAD method.

For each value of the surface index i, geometric data G_(i) derived from the surface patch S_(i,β) along with geometric data from the portion CP_(MS,β) of the model space curve C_(MS) may be used to construct up to two output patches. The two output patches would correspond to opposite sides of the model space curve. In some embodiments, the output patches are Bézier output patches, i.e., of Bézier form in both parametric directions. In other embodiments, the output patches are of Bézier form at least in the x^(i) direction, but may be of B-Spline or NURBS form in the y^(i) direction. In yet other embodiments, the output patches may be of B-Spline, NURBS, or T-spline form in both the x^(i) and y^(i) directions, in which case, the model space curve need not be refined to the level of Bézier portions.

The two output patches, denoted {tilde over (S)}_(iA,β) and {tilde over (S)}_(iβ,β), may be defined respectively on rectangular domains {tilde over (Ω)}_(β) ^(iA) and {tilde over (Ω)}_(β) ^(iB). However, the output patches {tilde over (S)}_(iA,β) and {tilde over (S)}_(iB,β) may be associated with two respective sub-domains of the extraction domain Ω_(β) ^(i) as follows. We may consider the extraction domain Ω_(β) ^(i) as being partitioned into two sub-domains Ω_(β) ^(iA) and that meet along the distance-minimized pullback of the model space curve C_(MS) into the 2D parametric domain Ω^(i) of the surface S₁. (The generic point C_(MS)(ν) on the model space curve C_(MS) could be mapped to a point in the 2D parametric domain of the surface S_(i) by minimizing the distance between S_(i)(s^(i), t^(i)) and the point C_(MS)(ν). This mapping may be referred to as a distance-minimized pullback.) As described below, we may extract geometric data G_(iA) corresponding to the surface patch S_(i,β) and the sub-domain and extract geometric data G_(iB) corresponding to the surface patch S_(i,β) and the sub-domain The output patch {tilde over (S)}_(iA,β) may be computed based on the geometric data G_(iA) and the geometric data from the portion CP_(MS,β) of the model space curve C_(MS). The output patch {tilde over (S)}_(iA,β) may be computed based on the geometric data G_(iB) and the geometric data from the portion CP_(MS,β).

For each value of the surface index i and for each kϵ{A, B}, the output patch {tilde over (S)}_(ik,β) may have the desirable property that a boundary isocurve of that output patch {tilde over (S)}_(ik,β) will map onto the portion CP_(MS,β) of the model space curve C_(MS). Thus, the output patches {{tilde over (S)}ik,βiϵ{0,1}; kϵ{A, B}} will all meet each other along the portion CP_(MS,β) with C⁰ continuity. (Here we assume there are two surfaces participating in the surface-surface intersection. However, the principles of the present invention naturally generalize to any number of participating surfaces.)

Furthermore, if the above-described reparameterization step is performed for each value of the surface index i, then for all νϵI_(β), iϵ{0,1} and kϵ{A, B}, the y^(i) isocurve of {tilde over (S)}_(ik,β) corresponding to the value {tilde over (x)}^(i)=ν will either start or end at the point C_(MS)(ν) on the model space curve. Thus, longitudinal isocurves in different output patches will meet at the same point on the model space curve if they correspond to the same coordinate value ν. Because this property holds for each portion CP_(MS,β) of the model space curve (i.e., each value of portion index β in the index set Φ, the parameter spaces of the output patches

{tilde over (S)} _(ik,β) :iϵ{0,1,kϵ{A,B},βϵΦ}

are unified into a global parameter space.

(Here we again assume there are two surfaces participating in the surface-surface intersection. However, the principles of the present invention naturally generalize to any number of participating surfaces.)

Yet further, the boundary isocurve of {tilde over (S)}_(ik,β) for the boundary remote from the portion CP_(MS,β) may have the same trajectory in model space as the corresponding boundary isocurve of the surface patch S_(i,β).

The construction of each output patch {tilde over (S)}_(iA,β) and {tilde over (S)}_(iB,β) may involve the solution of a corresponding linear system of equations (or a corresponding collection of linear systems). In some embodiments, the coefficients of the linear system are known in advance, and thus, the inverse matrix for this linear system may be precomputed and hardcoded as part of a software and/or hardware implementation of the Post-SSI algorithm, for more efficient solution of the linear system.

In cases where the boundary representation is modeling a 2-manifold object, the Post-SSI algorithm may construct only one of the output patches {tilde over (S)}_(iA,β) and {tilde over (S)}_(iB,β) for each value of the surface index i.

In cases where the boundary representation is modeling a non-manifold object, the Post-SSI algorithm may construct two or more output patches that meet along the model space curve with at least C⁰ continuity. For example, the Post-SSI algorithm may construct both the output patches {tilde over (S)}_(iA,β) and {tilde over (S)}_(iB,β).

In some embodiments, the two output patches {tilde over (S)}_(iA,β) and {tilde over (S)}_(iB,β) may be combined (by knot removal) along the respective boundaries that map to the model space curve, in order to form a single output patch {tilde over (S)}_(i,β), in which case the two identified boundaries become an interior isoparametric trim curve for the single output patch {tilde over (S)}_(i,β).

In some embodiments, the above mentioned geometric data G_(i) derived from the surface patch S_(i,β) may include:

-   -   the control points of one or both of the above-described         reparametrized transverse boundary isocurves;     -   the control points of boundary isocurves that extend in the         y^(i) direction; and     -   the control points of one or more non-boundary isocurves that         extend in the y^(i) direction.

The above mentioned geometric data from the portion CP_(MS,β) of the model space curve C_(MS) may include the control points of that portion.

Sampling Longitudinal Isoparametric Curves from the Surface Patch

The Post-SSI algorithm may compute curve control points for a number of y^(i) isocurves based on the surface control points of the surface patch S₀. (By definition, a y^(i) isocurve corresponds to a fixed value of x^(i) and extends in the y^(i) direction.) In particular, curve control points for two boundary isocurves may be computed, i.e., boundary isocurves corresponding respectively to the boundary values x^(i)=x_(β) ^(i) and x^(i)=x_(β+1) ^(i) of the extraction domain Ω_(β) ^(i). Furthermore, curve control points for two or more non-boundary isocurves may be computed. The two or more non-boundary isocurves correspond respectively to two or more distinct values of x^(i) in the interior of the closed interval [x_(β) ^(i),x₊₁ ^(i)]. In some embodiments, the number of the non-boundary isocurves may be equal to one less than the degree p_(x) _(i) of the surface patch S_(i,β) in the x^(i) direction (or the degree of the surface S_(i) in the x^(i) direction). For example, if p_(x) _(i) =3, there may be two non-boundary isocurves. If p_(x) _(i) =2, there may be only one non-boundary isocurve. The Greville abscissae may be desirable values of x^(i) or ν^(i) to use for computing the p_(x) _(i) −1 non-boundary isocurves. Each of the non-boundary isocurves may correspond to a respective one of the Greville abscissae. The sampling of the non-boundary isocurves at the Greville abscissae can be shown to produce a solution to the ensuing linear system that is optimal in various metric norms.

More generally, if we denote the surface control net of the surface patch S_(i,β) by

{P _(λ,ρ): 0≤λ≤m _(x) _(i) ,0≤ρ≤n _(y) _(i) },

with the index λ being associated with the parameter x^(i) and the index ρ being associated with the parameter y^(i), the number of the non-boundary isocurves may be set equal to one less than m_(x) _(i) .

Let

ISO_(x_(λ)^(i))(y^(i))

denote the y^(i) isocurve of the surface patch S_(i,β) at x_(λ) ^(i):

ISO _(x) _(λ) _(i) (y ^(i))=S _(i,β)(x _(λ) ^(i) ,y ^(i)).

Techniques for computing the curve control points of isocurves given the surface control points of a surface patch are well understood in the art of spline-based modeling.

In at least some embodiments, there may be m_(x) _(i) +1 longitudinal isocurves, corresponding respectively to the m_(x) _(i) +1 locations

x ₀ ^(i) <x ₁ ^(i) < . . . <x _(λ) ^(i) < . . . <x _(m) _(x) _(i−1) ^(i) <x _(m) _(x) _(i) ^(i),

with x₀ ^(i)==x_(β) ^(i) and x_(m) _(x) _(i) ^(i)=x_(β+1) ^(i).

Restore Degree of Isoparametric Curves, as Needed

In some embodiments, the Post-SSI algorithm may employ a conventional software tool for computing the control points of an isocurve based on the control points of a spline surface and a fixed value of a given surface parameter. For some conventional tools, the isocurve returned by the tool may be of lower degree than the degree of the surface in the relevant direction when the trajectory of the isocurve is simple enough to admit an exact representation with a lower degree curve. (For example, imagine a bicubic spline surface generating by lofting, in which case the isocurves along one parametric direction may be line segments.) This property of conventional tools, while useful in many contexts, is not so useful in the present context. For the sake of the surface reconstruction to be described below, we may require that the above-mentioned y^(i) isocurves to be of degree equal to the degree p_(y) _(i) of the surface patch S_(i,β) in the y^(i) direction. To the extent that a software tool returns an isocurve of lower degree, the Post-SSI algorithm may elevate the degree of the isocurve to p_(y) _(i) . Techniques for degree elevation are well known in the art of spline-based modeling.

Divide the y^(i) Isoparametric Curves Based on Model Space Curve

Each of the above described y^(i) isocurves of the surface patch S_(i,β) may be divided into two sub-isocurves based on the model space curve. In some embodiments, for each λϵ{0, 1, 2, . . . , m_(x) ^(i)}, the isocurve

ISO_(x_(λ)^(i))(y^(i))

may be divided into two sub-isocurves at a location y^(i)=y_(SSI,λ) determined based on the model space curve (or the portion CP_(MS,β) of the model space curve). In particular, the Post-SSI algorithm may compute y_(SSI,λ) as the value of the ν_(λ) parameter that achieves the closest approach of the isocurve

ISO_(x_(λ)^(i))(y^(i))

to the model space curve C_(MS)(ν). In other words, the Post-SSI algorithm may minimize the distance between

ISO_(x_(λ)^(i))(y^(i))

and C_(MS)(ν) over the space {(ν, y^(i))}. (The value ν^(k) that achieves the minimum distance may be used in the optional constraint-based point repositioning described below.) The isocurve

ISO_(x_(λ)^(i))(y^(i))

may be refined at y^(i)=y_(SSI,λ), to divide the isocurve into two sub-isocurves, denoted

ISO_(x_(λ)^(i))^(A)(y^(i))  and  ISO_(x_(λ)^(i))^(B)(y^(i)).

FIGS. 21A-21C show an example of the above-described isoparametric curves sampling and division based on the model space curve. FIG. 21A shows a spline surface S₀=S₀(s⁰,t⁰) intersected by spline surface S₁=S₁ (s¹,t¹), a parameter space curve C_(PS0) (u⁰) corresponding to spline surface S₀, and a model space curve C_(MS)(ν). FIG. 21B shows an example of a reconstruction domain (in dark grey) corresponding to a portion of the parameter space curve C_(PS0)(u⁰) of type s⁰. (The reconstruction domain is constrained from above by the distance-minimized pullback of the model space curve, C_(MS)(ν).) FIG. 21C shows the corresponding sub-isoparametric curves

{ISO _(x) _(λ) _(i) ^(A)(y ^(i) =t ⁰):x _(λ) ^(i) =s _(β) ₀ ⁰ ,s _(γ) ₀ ⁰ ,s _(γ) ₁ ⁰ ,s _(β) ₁ ⁰}

generated from the above-described sampling and curve division procedures.

Reparameterization of y^(i) Isoparametric Subcurves to a Common Knot Vector

A) Linear Reparameterization and Knot Cross-Seeding of the y^(i) Isoparametric Subcurves.

In some embodiments, it is possible that different sub-isocurves of the set

{ISO_(x_(λ)^(i))^(A)(y^(i)):  λ = 0, 1, 2, …  , m_(x^(i))}

have different numbers of knots. (The sub-isocurves are typically of different lengths, and thus, a longer sub-isocurve might intercept a larger number of transverse knot lines of the surface patch S_(i,β) than a shorter sub-isocurve.) The Post-SSI algorithm may address this issue as follows. Each sub-isocurve

ISO_(x_(λ)^(i))^(A)(y^(i)),

λϵ{0, 1, 2, . . . , m_(x) _(i) }, may be linearly reparametrized to a common parametric interval I_({tilde over (y)}) _(i) ^(iA)=[{tilde over (y)}_(min) ^(iA),{tilde over (y)}_(max) ^(iA)] in the y^(i) parameter, to obtain a corresponding linearly reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}.$

(There are a wide variety of ways to select {tilde over (y)}_(min) ^(iA) and {tilde over (y)}_(max) ^(iA). For example, in one embodiment, I_({tilde over (y)}) _(i) ^(iA)=[0,1]. In another embodiment, I_({tilde over (y)}) _(i) ^(iA)=[y_(LB) ^(i),{tilde over (y)}_(SSI) ^(i)], where {tilde over (y)}_(SSI) ^(i) is strictly between y_(LB) ^(i) and y_(UB) ^(i), e.g., the midpoint of y_(LB) ^(i) and y_(UB) ^(i).) Furthermore, for each λϵ{0, 1, 2, . . . , m_(x) _(i) }, the Post-SSI algorithm may identify any knots of the reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\mu}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

μϵ{0, 1, 2, . . . , m_(x) _(i) }, μ≠λ, that are not present in the knot vector of the reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

and add those identified knots to the knot vector of

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}.$

In other words, wherever one reparametrized sub-isocurve has a knot that is absent from the knot vector of a different reparametrized sub-isocurve, we may add that knot to the knot vector of the different reparametrized sub-isocurve. As a result, each of the reparametrized sub-isocurves will share a common knot vector.

FIG. 21D shows an example of the linear reparameterization based on the sub-isocurves shown in FIG. 21C.

B) Non-Linear Reparameterization of the y^(i) Isoparametric Subcurves.

In some embodiments, each sub-isocurve

ISO_(x_(λ)^(i))^(A)(y^(i)),

λϵ{0, 1, 2, . . . , m_(x) _(i) }, may be non-linearly reparametrized with a corresponding reparameterization function

${y^{i} = {f_{x_{k}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}},$

to obtain a corresponding reparametrized sub-isocurve

${{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

so that the reparametrized sub-isocurves

$\left\{ {{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)} \right\}$

are defined on a common parametric interval I_({tilde over (y)}) _(i) ^(iA)=[{tilde over (y)}_(min) ^(iA), {tilde over (y)}_(max) ^(iA)] and have a common knot vector. For example, the functions

{f_(x_(λ)^(i))^(A)}

may be 1D spline curves of degree p_(REPARAM), in which case the sub-isocurves

$\left\{ {{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)} \right\}$

will be of degree p_(y) _(i) p_(REPARAM), where p_(y) _(i) is the degree of the sub-isocurves

{ISO_(x_(λ)^(i))^(A)(y^(i))}.

(Non-linear reparameterization implies p_(REPARAM) is greater than one.)

In an alternative embodiment, the longest of the sub-isocurves and the remaining sub-isocurves may be addressed differently, but with the same goal of obtaining sub-isocurves defined on a common knot vector. In particular, one may first identify the longest of the sub-isocurves. Let x_(LONG) ^(iA) denote the value of x_(λ) ^(i) corresponding to the longest sub-isocurve. Each remaining sub-isocurve

ISO_(x_(λ)^(i))^(A)(y^(i)),

x_(λ) ^(i)≠x_(LONG) ^(iA), may then be non-linearly reparametrized with a corresponding reparameterization function

$y^{i} = {f_{x_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}$

of common degree p_(REPARAM), to obtain a corresponding non-linearly reparametrized sub-isocurve, denoted

${{EVRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

of degree p_(y) _(i) p_(Reparam) The longest sub-isocurve may then be degree elevated to degree p_(y) _(i) p_(Reparam), to obtain a degree-elevated sub-isocurve, denoted

${{EVRISO}_{{\overset{\sim}{x}}_{LONG}^{iA}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}.$

The nonlinear reparameterizations and the degree elevation are performed in such a fashion that the resulting set of m_({tilde over (x)}) _(i) +1 sub-isocurves

$\left\{ {{{{EVRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\text{:}\mspace{14mu} \lambda} \in \left\{ {0,1,2,\ldots,m_{{\overset{\sim}{x}}^{i}}} \right\}} \right\}$

are of degree p_(y) _(i) p_(Reparam) and share a common knot vector. (The prefix “EV” is meant as a reminder of the elevation of the longest sub-isocurve.) For example, if p_(y) _(i) =3, p_(Reparam)=3, and the sub-isocurve

ISO_(x_(m_(x^(i)))^(i))^(A)(y^(i))

is the longest of the sub-isocurves, then each of the sub-isocurves

ISO_(x_(λ)^(i))^(A)(y^(i)),

λϵ{1, 2, . . . , m_(x) _(i) } may be non-linearly reparametrized with a corresponding degree-3 reparameterization function

${y^{i} = {f_{x_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}},$

and the sun-isocurve

ISO_(x_(m_(x^(i)))^(i))^(A)(y^(i))

may be degree elevated to degree 9, to obtain a common knot vector. FIG. 21E shows an example of the above discussed non-linear reparameterization scheme, using the sub-isocurves of FIG. 21C.

In the following discussion of geometry reconstruction, it may not be significant how the sub-isocurves

{ISO_(x_(λ)^(i))^(A)(y^(i))}

have been reparametrized to a common knot vector. Thus, we may drop the prefix “L”, “NL” and “EV”, and refer simply to reparametrized sub-isocurves

$\left\{ {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)} \right\}.$

This notation is meant to encompass any type of reparameterization being applied to the sub-isocurves to obtain reparametrized sub-isocurves that share a common knot vector.

C) Selection of Domain for Construction of Output Patch {tilde over (S)}_(iA,β).

At this point, we may select the domain Ω_(β) ^(iA) for the output patch {tilde over (S)}_(iA,β). This domain may be interpreted as a reconstruction domain because the output patch {tilde over (S)}_(iA,β) may be constructed so as to rebuild (or modify) the geometry of the surface patch S_(i,β) on the extraction sub-domain Ω_(β) ^(iA).

In some embodiments, the reconstruction domain Ω_(β) ^(iA) may be a subregion of the extraction domain Ω_(β) ^(i). In some embodiments, the reconstruction domain Ω_(β) ^(iA) may be set equal to the rectangle

[{tilde over (x)} _(β) ^(i) ,{tilde over (x)} _(β+1) ^(i) ]×[{tilde over (y)} _(min) ^(iA) ,{tilde over (y)} _(max) ^(iA)],

where {tilde over (y)}_(min) ^(iA) and {tilde over (y)}_(max) ^(iA) are respectively the first and last knot values of the common knot vector of the reparametrized sub-isocurves

$\left\{ {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)} \right\},$

i.e., the minimum and maximum values of the common interval on which the reparametrized sub-isocurves are defined.

In other embodiments, the reconstruction domain Ω_(β) ^(iA) may be set equal to the rectangle

[{tilde over (x)} _(β) ^(i) ,{tilde over (x)} _(β+1) ^(i) ]×[{tilde over (y)} _(LIK) ^(iA) ,{tilde over (y)} _(max) ^(iA)],

where {tilde over (y)}_(max) ^(iA) is the last knot value of the common knot vector, and {tilde over (y)}_(LIK) ^(iA) is the last interior knot of the common knot vector. (An interior knot of a curve is a knot occurring in the interior of the domain of that curve.) Each reparametrized sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

λϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }, may be refined at {tilde over (y)}^(i)={tilde over (y)}_(LIK) ^(iA) to obtain a corresponding Bézier sub-isocurve

${BISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)$

on the domain [{tilde over (y)}_(LIK) ^(iA), {tilde over (y)}_(max) ^(iA)].

D) Counterpart Processing of Sub-Isocurves on Other Side of Model Space Curve.

Note that similar processing operations may be applied to the sub-isocurves on the other side of the model space curve, i.e., on the sub-isocurves

{ISO_(x_(λ)^(i))^(B)(y^(i)):  λ = 0, 1, 2, …, m_(x^(i))}.

In some embodiments, each sub-isocurve

ISO_(x_(λ)^(i))^(B)(y^(i)),

λϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }, may be linearly reparametrized to a common parametric interval I_({tilde over (y)}) _(i) ^(iB)=[{tilde over (y)}_(min) ^(iB),{tilde over (y)}_(max) ^(iB)] in the y^(i) parameter, to obtain a corresponding reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}.$

(There are a wide variety of ways to select {tilde over (y)}_(min) ^(iB) and {tilde over (y)}_(max) ^(iB). For example, in one embodiment, I_({tilde over (y)}) _(i) ^(iB)=[0,1]. In another embodiment, I_({tilde over (y)}) _(i) ^(iB)=[{tilde over (y)}_(SSI) ^(i),{tilde over (y)}_(UB) ^(i)], where {tilde over (y)}_(LB) ^(i)<{tilde over (y)}_(SSI) ^(i)<{tilde over (y)}_(UB) ^(i).) Furthermore, for each λϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }, the Post-SSI algorithm may identify any knots of the reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

μϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }, μ≠λ, that are not present in the knot vector of the reparametrized sub-isocurve

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

and add those identified knots to the knot vector of

${{LRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}.$

As a result, each of the reparametrized sub-isocurves will share a common knot vector.

In some embodiments, each sub-isocurve

ISO_(x_(λ)^(i))^(B)(y^(i)).

λϵ{0, 1, 2, . . . , m_(x) _(i) }, may be non-linearly reparametrized with a corresponding reparameterization function

${y^{i} = {f_{x_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}},$

to obtain a corresponding reparametrized sub-isocurve

${{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

so that the reparametrized sub-isocurves

$\left\{ {{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)} \right\}$

are defined on a common parametric interval I_({tilde over (y)}) _(i) ^(iB)=[{tilde over (y)}_(min) ^(iB),{tilde over (y)}_(max) ^(iB)] and have a common knot vector. For example, the functions

{f_(x_(λ)^(i))^(B)}

may be 1D spline curves of degree p_(Reparam), in which case the sub-isocurves

$\left\{ {{NLRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)} \right\}$

will be of degree p_(y) _(i) p_(REPARAM), where p_(y) _(i) is the degree of the sub-isocurves

{ISO_(x_(λ)^(i))^(B)(y^(i))}.

In an alternative embodiment, the longest of the sub-isocurves and the remaining sub-isocurves may be addressed differently, but with the same goal of obtaining sub-isocurves defined on a common knot vector. In particular, one may first identify the longest of the sub-isocurves Let x_(LONG) ^(iB) denote the value of x_(λ) ^(i) corresponding to the longest sub-isocurve. Each remaining sub-isocurve

ISO_(x_(λ)^(i))^(B)(y^(i)),

x_(λ) ^(i)≠x_(LONG) ^(iB), may then be non-linearly reparametrized with a corresponding reparameterization function

$y^{i} = {f_{x_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}$

of common degree p_(Reparam), to obtain a corresponding non-linearly reparametrized sub-isocurve, denoted

${{EVRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

of degree p_(y) _(i) p_(REPARAM). The longest sub-isocurve may then be degree elevated to degree p_(y) _(i) p_(REPARAM), to obtain a degree-elevated sub-isocurve

${{EVRISO}_{{\overset{\sim}{x}}_{LONG}^{iB}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}.$

The nonlinear reparameterizations and the degree elevation are performed in such a fashion that the resulting set of m_({tilde over (x)}) _(i) +1 sub-isocurves

$\left\{ {{{{EVRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}\text{:}\mspace{14mu} \lambda} \in \left\{ {0,1,2,\ldots,m_{{\overset{\sim}{x}}^{i}}} \right\}} \right\}$

are of degree p_(y) _(i) p_(REPARAM) and share a common knot vector.

In the following discussion of geometry reconstruction, it may not be significant how the sub-isocurves

{ISO_(x_(λ)^(i))^(B)(y^(i))}

have been reparametrized to a common knot vector. Thus, we may drop the prefix “L”, “NL” and “EV”, and refer simply to reparametrized sub-isocurves

$\left\{ {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)} \right\}.$

This notation is meant to encompass any type of reparameterization being applied to the sub-isocurves to obtain reparametrized sub-isocurves that share a common knot vector.

At this point, we may select the domain {tilde over (Ω)}_(β) ^(iB) for the output patch {tilde over (S)}_(iB,β). This domain may be interpreted as a reconstruction domain because the output patch {tilde over (S)}_(iB,β) may be constructed so as to rebuild (or modify) the geometry of the surface patch S_(i,β) on the extraction sub-domain {tilde over (Ω)}_(β) ^(iB).

In some embodiments, the reconstruction domain {tilde over (Ω)}_(β) ^(iB) may be a subregion of the extraction domain {tilde over (Ω)}_(β) ^(i). In some embodiments, the reconstruction domain {tilde over (Ω)}_(β) ^(iB) may be set equal to the rectangle

[{tilde over (x)} _(β) ^(i) ,{tilde over (x)} _(β+1) ^(i) ]×[{tilde over (y)} _(min) ^(iB) ,{tilde over (y)} _(max) ^(iB)],

where {tilde over (y)}_(min) ^(iB) and {tilde over (y)}_(max) ^(iB) are respectively the first and last knot values of the common knot vector of the reparametrized sub-isocurves

$\left\{ {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)} \right\},$

i.e., the minimum and maximum values of the common interval on which the reparametrized sub-isocurves are defined.

In other embodiments, the reconstruction domain {tilde over (Ω)}_(β) ^(iB) may be set equal to the rectangle

[{tilde over (x)} _(β) ^(i) ,{tilde over (x)} _(β+1) ^(i) ]×[{tilde over (y)} _(min) ^(iB) ,{tilde over (y)} _(FIK) ^(iB)],

where {tilde over (y)}_(min) ^(iB) is the first knot value of the common knot vector, and {tilde over (y)}_(FIK) ^(iB) is the first interior knot of the common knot vector. Each reparametrized sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

λϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }, may be refined at {tilde over (y)}^(i)={tilde over (y)}_(FIK) ^(iB) to obtain a corresponding Bézier sub-isocurve

${BISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)$

on the domain is [{tilde over (y)}_(min) ^(iB), {tilde over (y)}_(FIK) ^(iB)].

Optional: Constraint-Based Repositioning of the Control Points of the Isoparametric Subcurves

In some embodiments, for each λϵ{0, 1, 2, . . . , m_(x) _(i) }, the Post-SSI algorithm may perform constraint-based point repositioning on the controls points of the sub-isocurve

${RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)$

and the control points of the counterpart sub-isocurve

${RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)$

so that those sub-isocurves meet on the model space curve C_(MS) and achieve a specified degree of continuity at the meeting point on the model space curve C_(MS). (The desired degree of continuity may be specified by a user.)

Let {{tilde over (Q)}_(λ,ρ) ^(A): 0≤λ≤n_({tilde over (y)}) _(i) ^(A)} denote the set of control points of the sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}.$

(While this set of control points depends among other things on surface index i, we have not added that index to the control point notation, for simplicity of discussion.) Among the two end control points of the sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)},$

i.e., {tilde over (Q)}_(λ,0) ^(A) and

${\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{A}}^{A},$

note that

${\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{A}}^{A}$

is closest to the model space curve C_(MS).

Let {{tilde over (Q)}_(λ,ρ) ^(A): 0≤λ≤n_({tilde over (y)}) _(i) ^(A)} denote the set of control points of the sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}_{i} \right)}.$

(As above, we suppress the dependency on surface index, for ease of discussion) Among the two end control points of the sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},$

i.e., {tilde over (Q)}_(λ,0) ^(B) and

${\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{B}}^{B},$

note that {tilde over (Q)}_(λ,0) ^(B) is closest to the model space curve C_(MS). (Because the sub-isocurves are constructed from a surface patch of the original surface S_(i), the sub-isocurves generically miss the model space curve.)

To guarantee that the sub-isocurves

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\mspace{14mu} {and}\mspace{14mu} {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}$

meet on the model space curve C_(MS), one need only set both

${\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{A}}^{A}$

and {tilde over (Q)}_(λ,0) ^(B) equal to a selected point on the model space curve, e.g., the point C_(MS)(ν^(k)) determined above:

${\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{A}}^{A} = {{\overset{\sim}{Q}}_{\lambda,0}^{B} = {{C_{MS}\left( v^{k} \right)}.}}$

To achieve C¹ continuity between the sub-isocurves

${{{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\mspace{14mu} {and}\mspace{14mu} {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}},$

in addition to the constraint of meeting on the model space curve, one would reposition the control points

${\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - 1})}}^{A}$

and {tilde over (Q)}_(λ,1) ^(B) to match the derivatives

$\frac{{dRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}{dy}\mspace{14mu} {and}\mspace{14mu} \frac{{{dRISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)},}{dy}$

at the point of meeting. Furthermore, the repositioning of those points may be constrained so that the derivative vector at the point of meeting on the model space curve equals (within a specified tolerance) the derivative vector of the original sub-isocurves at their original point of meeting on surface S_(i). (Because the original sub-isocurves are generated by dividing a larger isocurve, they have the same derivatives at their point of meeting on the surface S_(i).)

To achieve C² continuity between the sub-isocurves

${{{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\mspace{14mu} {and}\mspace{14mu} {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}},$

in addition to the constraint of meeting on the model space curve, one would reposition the control points

${\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - 2})}}^{A},{\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - 1})}}^{A},$

{tilde over (Q)}_(λ,1) ^(B) and {tilde over (Q)}_(λ,2) ^(B). Furthermore, the repositioning of those points may be constrained so that the first and second derivatives at the point of meeting on the model space curve are equal (within a specified tolerance) to the respective derivatives of the original sub-isocurves at their original point of meeting on surface S_(i).

More generally, to achieve C^(n) continuity between the sub-isocurves

${{{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\mspace{14mu} {and}\mspace{14mu} {{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)}},$

in addition to the constraint of meeting on the model space curve, one would reposition the control points

$\left\{ {{\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - n})}}^{A},\ldots,{\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - 2})}}^{A},{\overset{\sim}{Q}}_{\lambda,{({n_{{\overset{\sim}{y}}^{i}}^{A} - 1})}}^{A},{\overset{\sim}{Q}}_{\lambda,1}^{B},{\overset{\sim}{Q}}_{\lambda,2}^{B},\ldots,{\overset{\sim}{Q}}_{\lambda,n}^{B}} \right\}.$

Furthermore, the repositioning of those points may be constrained so that the first L derivatives at the point of meeting on the model space curve are equal (within a specified tolerance) to the respective derivatives of the original sub-isocurves at their original point of meeting on surface S_(i). Techniques for performing constraint-based repositioning of curve control points on general spline curves are well understood and need not be explained to those of ordinary skill in the art of spline theory.

In some embodiments, the above described constraint-based point repositioning may be performed on the Bézier sub-isocurves

$\left\{ {{BISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)} \right\}$

and/or the Bézier sub-isocurves

$\left\{ {{BISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{B}\left( {\overset{\sim}{y}}^{i} \right)} \right\}.$

Construction of Output Patch {tilde over (S)}_(iA,β) and/or Output Patch {tilde over (S)}_(iB,β).

The Post-SSI algorithm may construct the surface control points for the output patch {tilde over (S)}_(iA,β) based on a set of isoparametric curve data including:

-   -   (a) the control points of the portion CP_(MS,β) of the model         space curve, which may be interpreted as the control points for         a boundary isocurve of the output patch {tilde over (S)}_(iA,β);         and     -   (b) the control points of the y^(i) sub-isocurves

{RISO_({tilde over (x)}) _(λ) _(i) ^(A)({tilde over (y)} ^(i)):λϵ{0,1,2, . . . ,m _({tilde over (x)}) _(i) }},

or a subset of those control points.

See FIGS. 22A-22C and FIGS. 23A-23E to accompany the definitions below.

In some embodiments, the set of isoparametric curve data may also include the control points of the boundary isocurve

ISO_(y_(LB)^(i))(x^(i)).

In some embodiments, the set of isoparametric curve data may include the interior control points of the y^(i) sub-isocurves

$\left\{ {{{{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}\text{:}\mspace{14mu} k} \in \left\{ {0,1,2,\ldots,m_{{\overset{\sim}{x}}^{i}}} \right\}} \right\},$

but exclude the end control points of those sub-isocurves. A control point of a spline curve is said to be an interior control point if it is not an end control point of that spline curve.

The output patch {tilde over (S)}_(iA,β) may be defined on the above described reconstruction domain {tilde over (Ω)}_(β) ^(iA). For specificity of discussion, suppose the reconstruction domain {tilde over (Ω)}_(β) ^(iA) is given by:

[{tilde over (x)} _(β) ^(i) ,{tilde over (x)} _(β+1) ^(i) ]×[{tilde over (y)} _(LB) ^(i) ,{tilde over (y)} _(SSI) ^(i)],

where {tilde over (y)}_(LB) ^(i)<{tilde over (y)}_(SSI) ^(i)<{tilde over (y)}_(UB) ^(i), where {tilde over (y)}_(LB) and {tilde over (y)}_(UB) are respectively the lower and upper bounds of the extraction domain Ω_(β) ^(i).

Let {{tilde over (P)}_(θ,ξ) ^(A): 0≤θ≤m_({tilde over (x)}) _(i) , 0≤ξ≤n_({tilde over (y)}) _(i) ^(A)} denote the surface control net of the output patch {tilde over (S)}_(iA,β), where n_({tilde over (y)}) _(i) ^(A)+1 is the number of curve control points in each sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{\lambda}^{i}}^{A}\left( {\overset{\sim}{y}}^{i} \right)}.$

In the special case where the sub-isocurves are Bézier sub-isocurves, n_({tilde over (y)}) _(i) ^(A)=p_({tilde over (y)}) _(i) , where p_({tilde over (y)}) _(i) is the degree of the Bézier sub-isocurves. Furthermore, in the special case where the surface patch {tilde over (S)}_(i,β) is of Bézier form in the {tilde over (x)}^(i) direction, m_({tilde over (x)}) ^(i)=p_({tilde over (x)}) _(i) , where p_({tilde over (x)}) _(i) is the degree in the {tilde over (x)}^(i) direction.

In the rest of this section, for ease of discussion, we will drop the surface index i from the parameters {tilde over (x)}^(i) and {tilde over (y)}^(i), the limit values m_(x) _(i) and n_({tilde over (y)}) _(i) ^(A), the sample location {tilde over (x)}_(λ) ^(i), and the degrees p_({tilde over (x)}) _(i) and p_({tilde over (y)}) _(i) .

We want the surface control points of the output patch {tilde over (S)}_(iA,β) to be computed so that the portion CP_(MS,β) of the model space curve C_(MS) is a boundary isocurve of the output patch {tilde over (S)}_(iA,β), i.e., the boundary isocurve along the boundary corresponding to {tilde over (y)}={tilde over (y)}_(SSI). This condition may be achieved by setting the boundary row

{{tilde over (P)} _(θ,n) _({tilde over (y)}) _(A) ^(A):0≤θ≤m _({tilde over (x)})}

of surface control points for the output patch equal to the control points of the portion CP_(MS,β). (It is a mathematical fact that the curve control points of a boundary isocurve of a tensor product spline are respectively identical to the surface control points of the tensor product spline along the same boundary.)

Similarly, in some embodiments, we may want the surface control points of the output patch {tilde over (S)}_(iA,β) to be computed so that the boundary isocurve ISO_(y) _(LB) (x) of S_(i,β) is a boundary isocurve of the output patch {tilde over (S)}_(iA,β) along the boundary {tilde over (y)}={tilde over (y)}_(LB). This condition may be achieved by setting the boundary row {{tilde over (P)}_(θ,0) ^(A):0≤θ≤m_({tilde over (x)})} of surface control points equal to the control points of the boundary isocurve ISO_(y) _(LB) (x).

For the boundary column {{tilde over (P)}_(0,ξ) ^(A):0≤ξ≤n_({tilde over (y)}) ^(A)} of surface control points of the output patch {tilde over (S)}_(iA,β), we may set the interior members of that column equal to the corresponding interior control points of the boundary sub-isocurve RISO_({tilde over (x)}) ₀ ^(A)({tilde over (y)}). (Recall that {tilde over (x)}₀={tilde over (x)}_(β).) The end members of this column are already addressed by the above described control point identifications.

For the boundary column {{tilde over (P)}_(m) _({tilde over (x)},) _(ξ) ^(A):0≤ξ≤n_({tilde over (y)}) ^(A)} of surface control points of the output patch {tilde over (S)}_(iA,β), we may set the interior members of that column equal to the corresponding interior control points of the boundary sub-isocurve

${{RISO}_{{\overset{\sim}{x}}_{m_{\overset{\sim}{x}}}}^{A}\left( \overset{\sim}{y} \right)}.$

(Recall that {tilde over (x)}_(m) _({tilde over (x)}) ={tilde over (x)}_(β+1).) The end members of this column are already addressed by the above described control point identifications.

Thus far we have accounted for all the boundary control points of the surface control net, leaving only the (m_({tilde over (x)})−1)(n_({tilde over (y)}) ^(A)−1) interior surface control points:

{{tilde over (P)} _(θ,ξ) ^(A):0≤θ≤m _({tilde over (x)},)0≤ξ≤n _({tilde over (y)}) ^(A)}

to be determined. The interior surface control points may be computed by solving a linear system based on the fundamental equation for computing the curve control points of an isocurve based on the surface control points of a tensor product spline. In the present context, note that the {tilde over (x)}={tilde over (x)}_(λ) isocurve of the output patch {tilde over (S)}_(iA,β) is given by:

$\begin{matrix} {{{\overset{\sim}{S}}_{{iA},\beta}\left( {{\overset{\sim}{x}}_{\lambda},\overset{\sim}{y}} \right)} = {\sum\limits_{\theta = 0}^{m_{\overset{\sim}{x}}}\; {\sum\limits_{\xi = 0}^{n_{\overset{\sim}{y}}^{A}}\; {{N_{\theta,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{N_{\xi,p_{\overset{\sim}{y}}}\left( \overset{\sim}{y} \right)}{\overset{\sim}{P}}_{\theta,\xi}^{A}}}}} \\ {{= {\sum\limits_{\xi = 0}^{n_{\overset{\sim}{y}}^{A}}\; {{N_{\xi,p_{\overset{\sim}{y}}}\left( \overset{\sim}{y} \right)}\left( {\sum\limits_{\theta = 0}^{m_{\overset{\sim}{x}}}\; {{N_{\theta,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{\overset{\sim}{P}}_{\theta,\xi}^{A}}} \right)}}},} \end{matrix}$

where

$N_{\theta,p_{\overset{\sim}{x}}}$

denotes the θ-th NURBS basis function of degree p_({tilde over (x)}), and

$N_{\xi,p_{\overset{\sim}{y}}}$

denotes the ξ-th NURBS basis function of degree p_({tilde over (y)}). (In the special case where the curve portion CP_(MS,β) of the model space curve is a Bézier portion, the NURBS function

$N_{\theta,p_{\overset{\sim}{x}}}$

may specialize to the Bézier basis function β_(θ,p) _({tilde over (x)}) of degree p_({tilde over (x)}).) Note, in some embodiments, we may interpret the control points {tilde over (P)}_(θ,ξ) ^(A) as being points in 4D space, and recover 3D points by projection.)

Observe that the curve control points {{tilde over (Q)}_(ξ) ^(k): 0≤ξ≤n_({tilde over (y)}) ^(A)} for this isocurve {tilde over (S)}_(iA,β) are given by:

${\overset{\sim}{Q}}_{\xi}^{k} = {\sum\limits_{\theta = 0}^{m_{\overset{\sim}{x}}}\; {{N_{\theta,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{{\overset{\sim}{P}}_{\theta,\xi}^{A}.}}}$

For each ξϵ{1, 2, . . . , n_({tilde over (y)}) ^(A)−1} and each λϵ{1, 2, . . . , m_({tilde over (x)})−1}, we can set {tilde over (Q)}_(ξ) ^(k) equal to {tilde over (Q)}_(λ,ρ) ^(A), i.e., the ρ^(th) control point of the sub-isocurve RISO_({tilde over (x)}) _(λ) ^(A):

${\overset{\sim}{Q}}_{\lambda,\rho}^{A} = {\sum\limits_{\theta = 0}^{m_{\overset{\sim}{x}}}\; {{N_{\theta,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{{\overset{\sim}{P}}_{\theta,\xi}^{A}.}}}$

This strategy is reasonable since both the sub-isocurve RISO_({tilde over (x)}) _(λ) ^(A)({tilde over (y)}) and the isocurve {tilde over (S)}iA,β(x_(k),y) are evaluated at {tilde over (x)}={tilde over (x)}_(λ).

In the above summation, we can separate out the terms (θ=0 and θ=m_({tilde over (x)})) containing known surface control points {tilde over (P)}_(0,ξ) ^(A) and {tilde over (P)}_(m) _({tilde over (x)}) _(,ξ) ^(A) from the terms (θ=1, 2, . . . , m_({tilde over (x)})−1) containing unknown surface control points, and obtain:

${{\overset{\sim}{Q}}_{\lambda,\rho}^{A} - {{N_{0,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{\overset{\sim}{P}}_{0,\xi}^{A}} - {{N_{m_{\overset{\sim}{x}},p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{\overset{\sim}{P}}_{m_{\overset{\sim}{x}},\xi}^{A}}} = {\sum\limits_{\theta = 0}^{m_{\overset{\sim}{x}} - 1}\; {{N_{\theta,p_{\overset{\sim}{x}}}\left( {\overset{\sim}{x}}_{\lambda} \right)}{\overset{\sim}{P}}_{\theta,\xi}^{A}}}$

Thus, for each ρϵ{1, 2, . . . , n{tilde over (y)}^(A)−1} we have a corresponding determinate system of (m_({tilde over (x)}−)1) linear vector equations in (m_({tilde over (x)})−1) vector unknowns, which may be solved, e.g., using any standard technique from numerical linear algebra, to determine the interior surface control points in the ξ-th row of the surface control net {{tilde over (P)}_(θ,ξ) ^(A)}, i.e., the surface control points

{{tilde over (P)} _(θ,ξ) ^(A):0<θ<m _({tilde over (x)})}.

In other words, each interior row of the surface control net corresponds to a separate linear system, which may easily be solved to determine the interior members of that row. Thus, the entire set of interior surface control points of the output patch {tilde over (S)}_(iA,β) are readily obtained.

To our knowledge, this technique of solving for surface control points based on isocurve control points is a new contribution to the theory of tensor product splines.

In some embodiments, the above described constraint-based point repositioning is applied to the control points of the sub-isocurves {RISO_({tilde over (x)}) _(λ) ^(A)({tilde over (y)}))} before they are used to compute the surface control net of the output patch.

In some embodiments, we assume that at least the end control points

$\left\{ {{{\overset{\sim}{Q}}_{\lambda,n_{{\overset{\sim}{y}}^{i}}^{A}}^{A}\text{:}\mspace{14mu} 0} \leq \lambda \leq n_{{\overset{\sim}{y}}^{i}}^{A}} \right\}$

of the sub-isocurves {RISO_({tilde over (x)}) _(λ) ^(A)({tilde over (y)}):0≤λ≤n_({tilde over (y)}) _(i) ^(A)} have been repositioned so as to reside on the model space curve prior to computation of the surface control net of the output patch. In these embodiments, the λ=n_({tilde over (y)}) _(i) ^(A) row of the surface control net may be addressed by linear system solution, just like the arbitrary interior row of the surface control net. Similarly, the λ=0 row of the surface control net may be addressed by linear system solution, just like the arbitrary interior row of the surface control net. In some embodiments, all rows of the surface control net are addressed by linear system solution.

In some embodiments, the Post-SSI algorithm may similarly construct the surface control points for the output patch {tilde over (S)}_(iB,β) based on a set of isoparametric curve data including:

-   -   (a) the control points of the portion CP_(MS,β) of the model         space curve;

(b) the control points of the y^(i) sub-isocurves

{RISO_({tilde over (x)}) _(λ) ^(B)({tilde over (y)} ^(i)):λϵ{0,1,2, . . . ,m _({tilde over (x)}) _(i) }}

or a subset of those control points.

In some embodiments, the set of isoparametric curve data may also include the control points of the boundary isocurve

ISO_(y_(LB)^(i))(x^(i)).

In some embodiments, the set of isoparametric curve data may include the interior control points of the y^(i) sub-isocurves {RISO_({tilde over (x)}) _(λ) ^(B) ({tilde over (y)}^(i)):λϵ{0, 1, 2, . . . , m_({tilde over (x)}) _(i) }}, but exclude the end control points of those sub-isocurves. 

1. A computer-implemented method for modifying a computer-aided design (CAD) model of a tangible object, the method comprising: performing, by the computer: storing geometric input data describing first and second input parametric surfaces associated with the CAD model, wherein the first and second input surfaces are described in a first and second parameter space domain, respectively; storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve is a parametric curve that approximates a geometric intersection of the first and second input surfaces; reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve; constructing first and second output surfaces as approximations of at least portions of the first and second input surfaces, respectively, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve; and storing a modified CAD model of the tangible object comprising the first and second output surfaces.
 2. The method of claim 1, wherein the model space trim curve comprises sequential segments of isocurves in the third parameter space domain.
 3. The method of claim 1, wherein the model space trim curve comprises a knot vector and isocurve sampling points, wherein the isocurve sampling points are used to sample isocurves of the first and second input surfaces.
 4. The method of claim 3, wherein said reparametrizing comprises, for the sampled isocurves, constructing a tensor-product surface that maps the first and second parameter space domains to the third parameter space domain.
 5. The method of claim 1, wherein the first and second output surfaces are within a predetermined or user-definable error tolerance of the first and second input surfaces, respectively.
 6. The method of claim 1, wherein the model space trim curve is received as an output from a Boolean operation performed on the first and second input surfaces in a computer-aided design (CAD) system.
 7. The method of claim 1, the method further comprising: constructing the model space trim curve based on the characteristic points of the first and second input surfaces and the geometric intersection of the first and second input surfaces.
 8. The method of claim 7, the method further comprising: identifying and classifying segments of the model space trim curve based on the characteristic points, wherein each identified segment comprises an isocurve in the third parameter space domain, wherein said classification determines a parameter in the third parameter space domain that has a constant value along the isocurve.
 9. The method of claim 1, wherein the first and second input surfaces and the first and second output surfaces are non-uniform rational basis splines (NURBS).
 10. The method of claim 1, wherein said modifying is performed as part of a Boolean operation in a computer-aided design (CAD) software system.
 11. The method of claim 1, the method further comprising: prior to said reparametrizing the first and second parametric surfaces into the common third parameter space domain based on the model space trim curve: constructing intermediate trim curves comprised within at least one of the first and second input surfaces; and reparametrizing at least one of the first and second parametric surfaces based on the intermediate trim curves.
 12. The method of claim 1, the method further comprising: prior to said reparametrizing the first and second parametric surfaces into the third parameter space domain based on the model space trim curve: selecting a parameter change handling (PCH) algorithm, wherein the PCH algorithm is selected from a set of available PCH algorithms including a null algorithm, and embedded extensions algorithm, and an extraordinary point insertion algorithm; and reparametrizing the first and second parametric surfaces based on the selected algorithm.
 13. The method of claim 12, wherein the null algorithm is a pass-through algorithm that does nothing; wherein the embedded extensions algorithm performs tangent extension without introducing extraordinary points; wherein the extraordinary point insertion algorithm introduces an intermediate trim curve that intersects the model space trim curve and is orthogonal to the model space trim curve at the intersection.
 14. The method of claim 1, wherein said reparametrizing comprises constructing a basis spline function to map each of the first and second parameter space domains into the third parameter space domain.
 15. The method of claim 1, the method further comprising: executing an engineering analysis on the modified CAD model to obtain data predicting physical behavior of an object described by the CAD model.
 16. The method of claim 15, wherein the engineering analysis comprises an isogeometric analysis.
 17. The method of claim 1, the method further comprising: directing a process of manufacturing an object described by the modified CAD model.
 18. The method of claim 1, the method further comprising: generating an image of an object based on the modified CAD model.
 19. The method of claim 1, the method further comprising: generating a sequence of animation images based on the modified CAD model and displaying the sequence of animation images.
 20. A computer-implemented method for modifying computer-aided design (CAD) model of a three-dimensional object, the method comprising: performing, by the computer: storing the CAD model in a memory of the computer, wherein the CAD model comprises a plurality of input surfaces comprising first and second input surfaces that cross in space along at least one intersection curve shared by the first and second input surfaces, wherein the first and second input surfaces are parametric surfaces that are described in a first and second parameter space domain, respectively; identifying a series of points along the at least one intersection curve; constructing a series of isocurves based on the identified series of points, wherein the constructed series of isocurves define a third parameter space domain; and modifying the CAD model by: constructing first and second output surfaces by reparametrizing the first and second input surfaces into the third parameter space domain based on the constructed series of isocurves; and replacing, in the CAD model, the first and second input surfaces with the first and second output surfaces to produce a modified CAD model of the three-dimensional object.
 21. The method of claim 20, wherein the first and second output surfaces are within a predetermined or user-definable error tolerance of the first and second input surfaces.
 22. A computer-readable memory medium comprising program instructions which, when executed by a computer, are configured to modify a computer-aided design (CAD) model of a three-dimensional object by: storing geometric input data describing a first and second input surface associated with the CAD model, wherein the first and second input surfaces are parametric surfaces described in a first and second parameter space domain, respectively; storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve is a parametric curve that approximates a geometric intersection of the first and second input surfaces; reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve; constructing first and second output surfaces as approximations of at least portions of the first and second input surfaces, respectively, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve; and storing a modified CAD model of the three-dimensional object comprising the first and second output surfaces.
 23. The memory medium of claim 22, wherein said reparametrizing comprises constructing a spline function to map each of the first and second parameter space domains into the third parameter space domain.
 24. The memory medium of claim 22, wherein the model space trim curve comprises sequential segments of isocurves in the third parameter space domain.
 25. The memory medium of claim 22, wherein the model space trim curve comprises a knot vector and isocurve sampling points, wherein isocurve sampling points are used to sample isocurves of the first and second input surfaces, respectively.
 26. The memory medium of claim 25, wherein said reparametrizing comprises, for the sampled isocurves, constructing a tensor-product surface that maps the first and second parameter space domains to the third parameter space domain.
 27. The memory medium of claim 22, wherein the first and second output surfaces are within a predetermined or user-definable error tolerance of the first and second input surfaces, respectively.
 28. The memory medium of claim 22, wherein the model space trim curve is received as an output from a Boolean operation performed on the first and second input surfaces in a computer-aided design (CAD) system.
 29. The memory medium of claim 22, wherein said modifying the CAD model further comprises: constructing the model space trim curve based on the characteristic points of the first and second input surfaces and the geometric intersection of the first and second input surfaces.
 30. The memory medium of claim 29, wherein said modifying the CAD model further comprises: identifying and classifying segments of the model space trim curve based on the characteristic points, wherein each identified segment comprises an isocurve in the third parameter space domain, wherein said classification determines a parameter in the third parameter space domain that has a constant value along the isocurve.
 31. The memory medium of claim 22, wherein the first and second input surfaces and the first and second output surfaces are non-uniform rational basis splines (NURBS).
 32. The memory medium of claim 22, wherein said modifying is performed as part of a Boolean operation performed in a computer-aided design (CAD) software system.
 33. The memory medium of claim 22, wherein said modifying the CAD model further comprises: prior to said reparametrizing the first and second parametric surfaces into the third parameter space domain based on the model space trim curve: constructing intermediate trim curves comprised within at least one of the first and second input surfaces; and reparametrizing at least one of the first and second parameter space domains based on the intermediate trim curves.
 34. The memory medium of claim 22, wherein said modifying the CAD model further comprises: prior to said reparametrizing the first and second parametric surfaces into the third parameter space domain based on the model space trim curve: selecting a parameter change handling (PCH) algorithm, wherein the PCH algorithm is selected from a set of available PCH algorithms including a null algorithm, and embedded extensions algorithm, and an extraordinary point insertion algorithm; and reparametrizing the first and second parametric surfaces based on the selected algorithm.
 35. The memory medium of claim 34, wherein the null algorithm is a pass-through algorithm that does nothing; wherein the embedded extensions algorithm performs tangent extension without introducing extraordinary points; wherein the extraordinary point insertion algorithm introduces an intermediate trim curve that intersects the model space trim curve and is orthogonal to the model space trim curve at the intersection.
 36. The memory medium of claim 22, wherein said modifying the CAD model further comprises: executing an engineering analysis on the modified CAD model to obtain data predicting physical behavior of an object described by the model.
 37. The memory medium of claim 36, wherein the engineering analysis comprises an isogeometric analysis.
 38. The memory medium of claim 22, wherein said modifying the CAD model further comprises: directing a process of manufacturing an object described by the modified CAD model.
 39. The memory medium of claim 22, wherein said modifying the CAD model further comprises: generating an image of an object based on the modified CAD model.
 40. The memory medium of claim 22 wherein said modifying the CAD model further comprises: generating a sequence of animation images based on the modified CAD model and displaying the sequence of animation images.
 41. The method of claim 1, wherein the modified CAD model is a model of a physical component.
 42. The method of claim 1, the method further comprising: using the modified CAD model to construct the tangible object.
 43. A computer-implemented method for modifying a computer-aided design (CAD) model of a vehicle component, the method comprising: performing, by the computer: storing geometric input data describing first and second input parametric surfaces associated with the CAD model of the vehicle component, wherein the first and second input surfaces are described in a first and second parameter space domain, respectively; storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve is a parametric curve that approximates a geometric intersection of the first and second input surfaces; reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve; constructing first and second output surfaces as approximations of at least portions of the first and second input surfaces, respectively, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve; and storing a modified CAD model of the vehicle component comprising the first and second output surfaces; wherein the modified CAD model of the vehicle component is useable to design or produce the vehicle component.
 44. The method of claim 43, wherein the modified CAD model of the vehicle component is a model of an automobile part.
 45. The method of claim 43, wherein the modified CAD model of the vehicle component is a model of an airplane part.
 46. A computer-implemented method for modifying a computer-aided design (CAD) model of a three-dimensional object rendered on a display, the method comprising: performing, by the computer: storing geometric input data describing first and second input parametric surfaces associated with the CAD model of the three-dimensional object, wherein the first and second input surfaces are described in a first and second parameter space domain, respectively; storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve is a parametric curve that approximates a geometric intersection of the first and second input surfaces; reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve; constructing first and second output surfaces as approximations of at least portions of the first and second input surfaces, respectively, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve; and storing a modified CAD model of the three-dimensional object comprising the first and second output surfaces, wherein the modified CAD model is a watertight representation of the first and second output surfaces; displaying on the display an animation which includes a rendering of the three-dimensional object based on the modified CAD model, wherein the rendering of the three-dimensional object based on the modified CAD model has reduced gaps along the geometric intersection of the first and second output surfaces due to the watertight representation of the first and second output surfaces. 